professional blackjack

PROFESSIONAL

BLACKJACK

STANFORD WONGPi Vee Press

PROFESSIONAL

BLACKJACK

byStanford Wong

Pi Vee Press

copyright © 1975, 1977, 1980, 1981, 1994by Pi Yee Press

All rights reserved No part of this book may be reproduced or utilized in any form or by any means, electronicor mechanical, including photocopying, recording, or byany information storage and retrieval system, withoutpermission in writing from the publisher. Inquiriesshould be addressed to Pi Yee Press, 7910 Ivanhoe #34,La Jolla, CA 92037-4511.

ISBN 0-935926-21-6

Printed in the United States of America

4 5 6 7 8 9 10

4 PROFESSIONAL BLACKJACK

PREFACE

This 1994 edition of Professional Blackjack is amajor update and expansion. It covers many rulesvariations that the previous edition did not cover. Itcontains simulation results to replace the "best guesses"in previous editions. Appendixes C, D, and E are newmaterial in this edition.

Parts of the previous edition were removed andexpanded into another book, Black.jack Secrets. Themain part involved was the chapter entitled "How to WinWithout Getting Kicked Out." After you master cardcounting, you will want to read Blackjack Secrets tolearn how to get away with it in a casino.

Professional Blackjack is a reference book for cardcounters. It contains a virtually complete set of strategynumbers for the high-low counting system. The numbers in this book have been objectively derived on acomputer. This material has been thoroughly tested incasinos throughout the world.

Win rates herein are estimates based on simulations by Blackjack Count Analyzer. The win rates con-

5

tained in this book are based on simulations totalingmore than ten billion hands of blackjack. You can buythis software and reproduce almost any strategy recommendation and almost any win rate in this book.

Small parts of this material first appeared in one oranother of the newsletters: Stanford Wong s BlackjackNewsletter, Current Blackjack News, Blackjack World,and Nevada Blackjack. (Of those, only Current Black-jack News is still published.)

Thanks to the people who read pre-publicationcopies and whose suggestions have made this a betterbook: Anthony Curtis, Michael Dalton, Dave Douglas,Frank Polo, Donald Schlesinger, and John Speer.

This book can be improved. If you find passages thatare wrong, or if your questions go unanswered, pleaseput your comments in writing and email them to me [email protected] or mail them to me at Pi Vee Press,7910 Ivanhoe #34, La Jolla, California 92037-4511. Ireserve the right to publish your questions and myanswers.


CONTENTS

CHAPTER 1 INTRODUCTION 15Dealer Action on Soft 17.................................................. 16"Pack" and "Deck" 16Multiple Numbers For Some Decisions 17Win Rates 17Benchmark Rules 18

CHAPTER 2 GENERIC BASIC STRATEGY 22Calculating Basic Strategy 23Pairs 24Soft Hands 25

Table 1 Generic Basic Strategy 26Hard Hands 2 8Surrender 28Insurance 29

CHAPTER 3 THE HIGH-LOW SYSTEM 30High-Low Count 3 1

Table 2 High-Low Count 3 1Count Per Deck 3 2Estimating Your Edge 34

Table 3 Player Edge 3 5

CONTENTS 7

Jumping into a Game in Progress 36Effect of Rules on Strategy 3 6Insurance 3 7Pairs 38

Table 4 Split, With No Double After, S17 3 8Table 5 Split, With Double After, S17 3 9

Doubling Down 40Table 6 Double Down, S17 4 1

Soft Standing 42Table 7 Soft Standing, S17 42

Hard Standing 43Table 8 Hard Standing, S17 43

If the Dealer Hits Soft Seventeen 44Summary Strategy Tables 45Importance of Strategy Tables 45

Table 9 Strategy Numbers, S17 46Table 10 Strategy Numbers, HI7 48

CHAPTER 4 INSURANCE 52Table 11 When to Take Insurance 53

A 10-Count for Insurance 54Table 12 10-Count for Insurance 54

Bustout Bet 56

CHAPTER 5 DOUBLE DOWN 58Double on Any Number of Cards 58Double on Two or Three Cards 59With Multiple-Card Surrender 59

Table 13 Double Down, S17 60Table 14 Double Down, H17 6 1Table 15 Double, Not Last Chance, S17 62Table 16 Double, Not Last Chance, HI7 63Table 17 Double Down, With Multiple-Card

Surrender, S17 64Table 18 Double Down, With Surrender After

Doubling, S17 65With Surrender After Doubling 66


CHAPTER 6 SPLIT 67Resplits 68Double Any Number of Cards 68

Table 19 Split, With No Double After, SI7 70Table 20 Split, With Double After, S17 7 1Table 21 Split, With No Double After, H17 72Table 22 Split, With Double After, HI7 73Table 23 Split, With Double Any Number of Cards

and No Double After Split, S17 74Table 24 Split, With Double Any Number of Cards

and No Double After Split, HI7 7 5Table 25 Split, With Double Any Number of Cards

Including After Split, S17 76Table 26 Split, With Double Any Number of Cards

Including After Split, HI7 77Table 27 Split, With Double Any Two or Three

Cards Before Split and Any Two After, S17 7 8Table 28 Split, With Double Any Two or Three

Cards Before Split and Any Two After, HI7 79

CHAPTER 7 NO HOLE CARD 80Lose Only Original Bet to a Natural 81Lose All to a Natural 8 1

Table 29 Lose All to a Natural, S17 82Any 21 Ties Dealer Natural 85

Table 30 Modifications to Appendixes When YouLose All to a Natural 8 5

Table 31 No Dealer Natural, S17 86

CHAPTER 8 SURRENDER 89Early Surrender 90

Table 32 Early Surrender 9 1Late Surrender 92

Table 33 Late Surrender 93Table 34 Multiple-Card Early Surrender 94

Multiple-Card Surrender 94Table 35 Multiple-Card Late Surrender 95

Surrender After Doubling Down 96

CONTENTS 9

Second-Chance 21 96Table 36 Second-Chance 21 97

CHAPTER 9 FIVE CARDS 98Half a Bet 99

Table 37 2-Card Hands With 5-Card Half Win,817 100

Table 38 3-Card Hands With 5-Card Half Win,8 17 102



Table 41 2-Card Hands With 5-Card Early HalfWin, 817 106

Table 42 3-Card Hands With 5-Card Early HalfWin, 8 17 108



Table 45 3-Card Hands With 6-Card AutomaticWin, 8 I7 111

Table 46 4-Card Hands With 6-Card AutomaticWin, 817 112

Table 47 5-Card Hands With 6-Card AutomaticWin, 817 11 3

Table 48 2-Card Hands With db3, 6-Card Win,HI 7 114

Table 49 3-Card Hands With db3, 6-Card Win,HI 7 116

Table 50 4-Card Hands With 6-Card Win, HI7 .. 117Table 51 5-Card Hands With 6-Card Win, HI7 .. II 8

Automatic Win 1I 9Table 52 2-Card Hands With Winning 5-Card 21

Pays 2-1, 817 120


Table 53 3-Card Hands With Winning 5-Card 21Pays 2-1, 817 122

Table 54 4-Card Hands With Winning 5-Card 21Pays 2-1, S17 123

Double Pay on Five-Card 21 124

CHAPTER 10 6-7-8 ETC 1256-7-8 126Double Pay for 6-7-8 126

Table 55 Any 6-7-8 Pays Double, SI7 127Table 56 6-7-8 Suited Pays Double, SI7 129

Triple Pay For 6-7-8 130Table 57 Any 6-7-8 Pays Triple, 817 131Table 58 6-7-8 Suited Pays Triple, S17 133Table 59 Any 6-7-8 Pays Double, H17 134Table 60 6-7-8 Suited Pays Double, H17 135Table 61 Any 6-7-8 Pays Triple, HI7 136Table 62 6-7-8 Suited Pays Triple, H17 13 7

7-7-7 138Table 63 7-7-7 Pays Double, 817 138Table 64 7-7-7 Pays Triple, S17 139Table 65 7-7-7 Pays Double, HI7 140Table 66 7-7-7 Pays Triple, HI7 141

CHAPTER 11 OVER/UNDER 13 142Basic Strategy 142Single Deck 143Counting Cards 145

Table 67 Crush Count 146

CHAPTER 12 HALVES COUNT 148Table 68 Comparison of High-Low and Halves

Counts 149Table 69 Halves Strategy Numbers, S17 152Table 70 Halves Strategy Numbers, H17 154

CHAPTER 13 DOUBLE EXPOSURE 156Estimating Your Edge 157

Table 71 Player Edge for Dbl Exposure 157

CONTENTS 11

Basic Strategy 158Counting Cards 159

Table 72 Double Exposure Basic, Part 1 160Table 73 Double Exposure Basic, 817 162Table 74 Double Exposure Basic, H17 164Table 75 High-Low, ObI Exposure, Part 1 168Table 76 High-Low, ObI Exposure, S17 170Table 77 High-Low, ObI Exposure, H17 172

Win Rates at Double Exposure 175Table 78 Halves, Dbl Exposure, Part 1 176Table 79 Halves, Double Exposure, S17 178Table 80 Halves, Double Exposure, HI7 180Table 81 Double Exposure Win Rates 182

CHAPTER 14 FINER POINTS 183Expected Win 183

Table 82 Effects on Win Rates 185Risk 189

Table 83 Area in the Tail of a NormalDistribution 191

Table 84 Probabilities of Low Points 197Chance of Reaching All-Time High 200Avoid Going Broke: Use Proportional Betting 201Optimal Bet Size 201

Table 85 Variance and Covariance for Blackjack 203Table 86 Optimal Bet as a Proportion of Your

Advantage 204Overbetting 206Stopping Rules 207The Optimal Number of Simultaneous Hands 208All Decisions Are Approximations 2 11Which Counting System 213How Many Numbers To Learn 215Side Count of Aces 2 I 7Two-Card Combinations 21 7Resolving Toss-ups 218Quick and Accurate 2 18


Thorp's Complete Point Count 219With Other People at the Table 220

CHAPTER 15 SHUFFLES 222New Decks Shuffled, But Not Together 223Single-Deck Shuffle Study 224

Table 87 Raw Data From Ten Shuffles 226Table 88 Card Positions of Ten Shuffles 227Table 89 Post-Shuffle Gaps Between Initially

Adjacent Cards 229Speed of Blackjack 23 1

Table 90 Average Times for Shuffling and DealingSingle-Deck Blackjack In Reno 232

Table 91 Average Rounds Per Hour for Single-DeckBlackjack in Reno 234

Table 92 Average Times for Shuffling and DealingSix-Deck Blackjack in Atlantic City 235

Table 93 Average Rounds Per Hour for Six-DeckBlackjack in Atlantic City 23 7

Search For Streakiness 239Table 94 Outcomes Versus Results of Previous Two

Hands 241Table 95 Average Probabilities for Dealer's Hand,

SI7 242Table 96 Dealer's Hand Versus Two Preceding

Dealer Hands 243Table 97 One Player, $100 Bets, Six Unshuffled

Decks, Lay and Pay 247Table 98 Any Two Cards from Well-Shuftled Six-

Deck Shoes 249Table 99 Consecutive Cards From Unshuffled Six-

Deck Shoes 250Table 100 Alternate Cards From Unshuffled Six-

Deck Shoes 251

APPENDIXES A & B 252Appendix A 254

Table Al High-Low, One Deck, HI7 255

CONTENTS 13

Table A2 High-Low, Four Decks, H17 259Table A3 High-Low, One Deck, 817 263Table A4 High-Low, Four Decks, 817 267

Appendix B 252Table B1 Halves, One Deck, H17 253Table B2 Halves, Four Decks, HI7 257Table B3 Halves, One Deck, S17 261Table B4 Halves, Four Decks, SI7 265

APPENDIX C 286Table C 1 Frequencies and Win Rates 6-Deck

Benchmark 288Table C2 Frequencies and Win Rates 2-Deck

Benchmark 288Table C3 Frequencies and Win Rates I-Deck

Benchmark 288Table C4 Frequencies and Win Rates 6-Deck,

Shuffle at 62 289Table C5 Frequencies and Win Rates 6-Deck,

Shuffle at 42 289Table C6 Frequencies and Win Rates Halves, 6-

Deck 289Table C7 Frequencies and Win Rates Double

Exposure Benchmark 290Table C8 Frequencies and Win Rates Double

Exposure Halves 290

APPENDIX D 291Table Dl Frequencies (per 100,000) of Initial Hands,

1 Deck, 817 294Table D2 Frequencies (per 100,000) of Decisions, 1

Deck, S17 295Table D3 Frequencies (per 100,000) of Initial Hands,

6 Decks, S17 296Table D4 Frequencies (per 100,000) of Decisions, 6

Decks, S17 297


APPENDIX E 2981 Deck, S17, Dealer Shows Ace 3021 Deck, H17, Dealer Shows Ace 3031 Deck, S17, Dealer Shows 2 3041 Deck, H17, Dealer Shows 2 3051 Deck, S17, Dealer Shows 3 3061 Deck, H17, Dealer Shows 3 3071 Deck, S17, Dealer Shows 4 3081 Deck, HI7, Dealer Shows 4 3091 Deck, S17, Dealer Shows 5 3101 Deck, H17, Dealer Shows 5 3111 Deck, S17, Dealer Shows 6 3121 Deck, HI7, Dealer Shows 6 3131 Deck, Dealer Shows 7 3141 Deck, Dealer Shows 8 3151 Deck, Dealer Shows 9 3161 Deck, Dealer Shows 10 3 176 Decks, S17, Dealer Shows Ace 3186 Decks, HI7, Dealer Shows Ace 3196 Decks, S17, Dealer Shows 2 3206 Decks, H17, Dealer Shows 2 32 16 Decks, S17, Dealer Shows 3 3226 Decks, H17, Dealer Shows 3 3236 Decks, S17, Dealer Shows 4 3246 Decks, H17, Dealer Shows 4 3256 Decks, S17, Dealer Shows 5 3266 Decks, H17, Dealer Shows 5 3276 Decks, S17, Dealer Shows 6 3286 Decks, HI?, Dealer Shows 6 3296 Decks, Dealer Shows 7 33 06 Decks, Dealer Shows 8 33 16 Decks, Dealer Shows 9 3326 Decks, Dealer Shows 10 333

GLOSSARy 334

SELECTED REFERENCES 342

INDEX 346

PUBLICATIONS BY STANFORD WONG. 349

15

CHAPTER 1

INTRODUCTION

The purpose of this book is to serve as a handyreference to card counters. When you travel, if there isany chance you will encounter a blackjack game, takethis book with you. Whatever rules you encounter atblackjack, you should be able to open this book and findthe strategy numbers you need to attack the game.

Chapter 2 presents generic basic strategy. Chapter3 presents the high-low counting system for one commonly-found set of rules. Following that are chapterspresenting strategy indexes for insurance, double down,splitting, the no-hole-card game, surrender, multiplecard bonuses, and bonuses for particular hands such as6-7-8 of the same suit. You ought to be able to find thestrategy index numbers you need for the rules you facein the casino of your choice.


Chapter 11 is devoted to over/under 13, a recentinnovation. Over/under 13 is two side bets. You can betthat your first two cards will total over thirteen or underthirteen. Aces count as one and not eleven, and thirteenloses. Over/under 13 can be profitably exploited by cardcounters.

Chapter 12 presents a more powerful countingsystem, the halves.

Chapter 13 is devoted to double exposure. This is aform of blackjack in which both the dealer's cards areexposed before the players act, and ties go to the dealer.

Chapter 14 explains some of the finer points ofblackjack, including expected win, risk, optimal betting,and counting in a casino.

Chapter 15 presents several studies of casinoshuffles.

The appendix contains tables of strategy indexes,expectations, and frequencies of particular hands.

Dealer Action on Soft 17Some of the tables in this book have been derived for

dealer stands on soft seventeen, and some have beenderived for dealer hits soft seventeen. An abbreviation isused to distinguish between the two: s17 means dealerstands on soft seventeen, and hI7 means dealer hits softseventeen.

"Pack" and "Deck"This book uses "deck" to mean 52 cards, and "pack"

to mean a collection of cards that might or might notcontain exactly 52 cards. The reason for this usage is to

INTRODUCTION 17

make the advice on card counting unambiguous: "countper deck" means "count per 52 cards."

Multiple Numbers For SomeDecisions

Some decisions have more than one number associated with them. For example, sometimes with 4-4 it iscorrect to hit at very low counts, double down at veryhigh counts, and split at intermediate counts. Wherethis complication exists, adequate explanation is provided in the text.

Another example: With soft eighteen you may wantto stand, you may want to hit, or you may want to doubledown. This complication generally is handled by havingtwo lines in the table, one for stand versus double andone for hit versus stand.

Win RatesThis book contains numerous win rates. They are

based on computer simulations of at least 30 millionhands of blackjack each. Their purpose is to aid inunderstanding the importance of various rules options.

Benchmark RulesI have selected a benchmark, an arbitrary set of

rules, playing conditions, and bets. This benchmark isshown on the next page. Use of a different benchmarkwould result in different numbers. However, the relativeimportance of different rules variations would be approximately the same no matter what set of rules wereused as the benchmark.

The benchmark betting scheme is not the best wayto bet. Much better is to leave the table on negative


Benchmark RulesSix decks.

One-deck cut, meaning five decks are dealt out.

Dealer stands on soft seventeen.

Double down is allowed on any first two cards butnot after splitting.

Resplits are allowed to a total of four hands. Splitaces receive one card each, and cannot be resplit.

Insurance is offered, but no other side bets.

No surrender.

Naturals pay 3:2, and tie a natural by the dealer.

The high-low counting system is used, with strat-egy numbers from -10 to +10.

There is a total of two players at the table.

The bet scheme is $100 at counts per deck of +4 ormore, $75 at +3, $50 at +2, $25 at zero or +1, and $10at all negative counts.

BenchmarkWin rate: $16Std dev: $415

INTRODUCTION 19

counts. On positive counts, you must bet in a mannerthat does not attract unwanted attention to yourself. SeeBlackjack Secrets for in-depth betting advice.

Simulating the base set of rules and betting schemefinds a win rate that is a benchmark for evaluating thecost or benefit of various deviations from the base rules.

Benchmark Win RateThe benchmark win rate is $16 per 100 hands.In a casino you will be able to play 50 to 300 hands

per hour, so 100 hands represents roughly an hour's play.Throughout this book results are stated in dollars perhour, which really means dollars per 100 hands.

Sampling ErrorThat $16 win rate is a simulation result, and thus is

an estimate. Though it is an estimate for 600 millionhands of blackjack, it still contains what statisticianscall "sampling error." That is, a different 600-millionhand simulation would result in almost but not quite thesame number. The term used to describe the precision ofan estimate is the standard error. The standard errorapplicable to the $16 win rate is $0.20. The actualsimulation result was $15.78, which was rounded because the pennies and dimes are not significant. So youshould think of the $16 as meaning somewhere in the$15.38 to $16.18 range. It would be nice to have a smallerstandard error of course, and the way to do that is toincrease the sample size. That is the reason why thesample size was run up to 600 million hands - to reducethe standard error to a reasonable number.

Each of the win rates in this book is based on asample of at least 30 million hands, and has a standarderror of less than $1. So if one rule is changed from thebenchmark, and a simulation with the new rule yields


a win rate of say $20 per hour, that rule increased the winrate by $4 per hour. That $4 can be thought of as thevalue per 100 hands to a card counter of that rulechange. But it is not exactly $4 per 100 hands becauseboth the $20 and the $16 are sample statistics withsampling error. If you run your own simulation trying toduplicate these results you probably would get $4, butyou could possibly get $3 or $5 as the value of the rule.

And of course if you use a different bet scheme, adifferent counting system, make an occasional error, orplay against a different number of decks or with adifferent number of players at the table, or with differentpenetration, the value to you might be different from $4per hour.

Average Bet SizeThe average bet size for the benchmark is $26.54.

That is the average initial bet; it does not includeadditional amounts wagered on doubles, splits, and sidebets such as insurance. That $26.54, plus or minus apenny or two, is the average bet size on all simulationsreported in this book for six decks, penetration down tothe last deck, and the high-low system.

Benchmark RiskThe standard deviation for 100 hands for the bench

mark is $415. That is, the win rate may be $16 but anygiven 100 hands typically has a result considerably lessthan or more than $16. In repeated samples of 100hands, 2/3 of the samples have wins of $16 plus or minus$415.

Risk varies with the rules. Some rule changes meanmore risk, and some mean less risk. Being able to doubledown more often (such as after splitting pairs) meansmore risk, as represented by a higher standard devia-

INTRODUCTION 21

tion. Restrictions on doubling down mean a lower winrate but that is partially offset by lower risk.

Win Rates ReportedA small box is used to indicate a win rate, and is the

result of a simulation of at least 30 million hands. Thebox explains the manner in which the rules of thissimulation differ from the benchmark, the win rate per100 hands using the benchmark betting scheme, andthe standard deviation applicable to 100 hands. The restof the information pertaining to that simulation is thesame as the benchmark as discussed above. As anexample, if one deck with penetration 26 cards is usedinstead of six decks, but everything else kept the sameas the benchmark, the win rate is $48 per 100 hands.

A summary of win rates is presented in table 82(page 185). Double-exposure win rates are presentedseparately, as table 81 (page 182).

One deckWin rate: $48Std dev: $482


CHAPTER 2

GENERICBASIC STRATEGY

This chapter contains generic basic strategy, and isa copy of chapter 2 of Basic Blackjack and BlackjackSecrets. If you already know basic strategy, skip thischapter.

Basic strategy is the best way to play a blackjackhand on the first round after a shuflle, assuming you seeno cards other than your own and the dealer's upcard.For a person who does not count cards, basic strategy isthe best way to play every hand.

This chapter presents basic strategy for single exposure, which is blackjack where the dealer has one cardface up for you to see as you are playing your hand. Basicstrategy is what plays you should make if you are notcounting cards and you do not have any informationabout the dealer's hole card. It is presumed that you

GENERIC BASIC STRATEGY 23

know the total in your own hand and the dealer's upcard,but no other cards.

Chapter 13 contains basic strategy for double exposure, which is blackjack where the dealer has two cardsface up.

Another book, Basic Blackjack, presents basicstrategy for all the various rules that have been offeredfor blackjack, such as dealers take ties on seventeen,six-card hand pays double, surrender after doublingdown, etc.

Calculating Basic StrategyBasic strategy can be either total-dependent or

composition-dependent. Total-dependent means thestrategy numbers require only the dealer's card and thetotal points in your hand. Composition-dependent meansthe strategy numbers require knowledge of the dealer'scard and the precise cards that make up your hand. Forexample, total-dependent strategy says stand on twelveagainst 4. Composition-dependent strategy for twelveagainst 4 requires you to specify how you get to twelve:Do you have 7-5, 8-4, 3-2-2-5, or what? If you get to twelveby 10-2 or 2-10 (where 10 means any 10-count card), andtwo or fewer decks are being used (or seven or fewer if thedealer stands on soft seventeen), you should hit. If youget to twelve by any other route, or enough decks arebeing used, you should stand. See Peter Griffin's TheTheory of Blackjack for a good discussion of compositiondependent strategy. (The 10-2 versus 4 advice is frompage 176 of Griffm's book.)

There are few differences between compositiondependent and total-dependent strategies for singledeck, and none that are important for multiple decks.


(The more decks shuffled together, the less differenceone card makes.) This book uses total-dependent strategy.

The details of basic strategy depend on the particulars of the rules. However, you must start someplace.This chapter presents a version of basic strategy that isapproximately correct for the most common sets of rules- a generic basic strategy. For modifications of thisgeneric basic strategy that are appropriate for differentsets of rules, see Basic Blackjack.

Blackjack is most commonly played with the dealer'shand showing one card face up. If you are playingblackjack in a game where you get to see two cards faceup in front of the dealer before you play your hand, go tochapter 13 for playing-strategy advice.

Table 1 presents generic basic strategy. It containsadvice for every decision the blackjack player commonlymakes. Each column is a different dealer upcard. (10, J,Q, and K are lumped together as 10.) Each row is adifferent player hand.

Technically, table 1 is basic strategy for multipledecks and dealer stands on soft seventeen.

PairsThe order of decisions presented in table 1 is the

order in which you evaluate your hand.Do you have a pair? At most casinos, any two 10

count cards, e.g. J-K, are a pair and may be split. If youhave a pair, the first part of table 1 tells you how to playyour hand. Use this part of the table to decide whetherto split your pair. To split means to make another betequal in size to your fIrst bet, and play each card as thestart of a separate hand.


If you split a pair and catch another card of the samevalue, resplit if you can. If it is correct to split a pair, it iscorrect to resplit.

You mayor may not be allowed to double down aftersplitting a pair. For example, if you split 8-8 and catch a3 for eleven, you mayor may not be allowed to doubledown on that eleven. If doubling down after splitting isallowed, then splitting is more attractive and you shouldsplit more often. The first part of table 1 assumes you arenot allowed to double down after splitting. If you areallowed to double down after splitting, then use the lastpart of table 1 - the part on the facing page. Thedecisions that are hits in the first part of table 1 andsplits on the facing page are: 6-6 against 2, 4-4 against5 or 6, 3-3 against 2 or 3, and 2-2 against 2 or 3.

Soft HandsDo you have an ace? Aces count your choice of either

eleven or one. A hand in which an ace counts eleven iscalled a soft hand, and the total points in it are called asoft total. The second part of table 1 explains how to playsoft hands.

The double-down advice is broken down into db anddbs. The reason is you need to know what to do with aparticular total if you cannot double down. For example,suppose you have soft eighteen and the dealer shows 3.Your best play is to double down, so that is what you doif you can. But if your soft eighteen is a three-card hand,say ace-2-5, then you probably will not be allowed todouble down. Table 1 lists "dbs" for that hand, whichmeans if you are not allowed to double down then youshould stand.


Table 1Generic Basic Strategy

Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 A

double not allowed after splitA-A spl spl spl spl spI spl spl spl spl spl10-109-9 spl spl spl spl spl spl spl8-8 spl spl spl spl spl spl spl spl spl spl7-7 spl spl spl spl spl spl h h h h6-6 h spl spl spl spl h h h h h5-5 db db db db db db db db h h4-4 h h h h h h h h h h3-3 h h spl spl spl spl h h h h2-2 h h spl spl spl spl h h h hsoft 21soft 20soft 19soft 18 dbs dbs dbs dbs h h hsoft 17 h db db db db h h h h hsoft 16 h h db db db h h h h hsoft 15 h h db db db h h h h hsoft 14 h h h db db h h h h hsoft 13 h h h h db h h h h hhard 21hard 20hard 19hard 18hard 17hard 16 h h sr sr srhard 15 h h h sr hhard 14 h h h h hhard 13 h h h h hhard 12 h h h h h h h

11 db db db db db db db db db h10 db db db db db db db db h h9 h db db db db h h h h h8 h h h h h h h h h h7 h h h h h h h h h h6 h h h h h h h h h h5 h h h h h h h h h h


Table 1 ContinuedPlayer's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 A

double allowed after splitA-A spl spl spl spl spl spl spl spl spl spl10-109-9 spl spl spl spl spl spl spl8-8 spl spl spl spl spl spl spl spl spl spl7-7 spl spl spl spl spl spl h h h h6-6 spl spl spl spl spl h h h h h5-5 db db db db db db db db h h4-4 h h h spl spl h h h h h3-3 spl spl spl spl spl spl h h h h2-2 spl spl spl spl spl spl h h h h

KEY

Stand.db Double down; if you cannot double, then hit.dbs Double down; if you cannot double, then stand.h Hit.

spl Split.s r Surrender; if you cannot surrender, then hit.


Note that sometimes it is correct to hit eighteen. Ifyou have soft eighteen and the dealer shows 9, 10, or ace,then hitting your soft eighteen is better than standingon it.

One decision that is very close is soft thirteenagainst 5. It does not matter whether you hit or doubledown.

Hard HandsThe lower two parts of table 1 explain how to play

the rest of your hands~ Hands labeled "hard" mightcontain aces, but under the circumstances all such acesare counted as one. Hands tabulated as from 5 to 11 donot contain an ace; if a hand totaling eleven or less hasan ace it is a soft hand and is played according to the"soft" part of the table.

SurrenderSurrender means losing half a bet for the privilege

of not playing out the hand. Late surrender means youcannot surrender if the dealer has a natural. The tableof generic basic strategy includes strategy for late surrender. Most of the' value of late surrender comes fromsurrendering sixteen against 10. Late surrender isworth 0.1% to a basic-strategy player.

If you are playing blackjack at a casino that does notoffer surrender, or if you are not allowed to surrenderdue to having more than two cards, then hit those handsfor which table 1 advises surrender.

GENERIC BASIC STRATEGY

Insurance

29

Table 1 does not show insurance. Basic strategy saysnever take insurance. Even if you have a natural, youare better off not insuring it. You are better off winning3:2 most of the time than winning even money for sure.


CHAPTER 3

THE HIGH-LOW

SYSTEM

How many dealer upcards can you see before youplay your hand? If you see two, go to chapter 12, whichdiscusses double exposure. This chapter discusses singleexposure blackjack.

Winning at blackjack requires two things: You mustbet more when you have the advantage and less whenthe dealer has the advantage; and you must makecorrect decisions on insurance, surrender, splitting pairs,doubling down, and hitting or standing.

Parts of this chapter are similar to chapter 3 ofBlackjack Secrets. One difference is that this book gives"surrender" its own chapter. Another difference is thatthis book presents high-low strategy numbers for a

HIGH-LOW SYSTEM 31

wider range of counts per deck, -10 to +10, instead of -1to +6.

High-Low CountYou need a counting system to tell whether you have

the advantage and to aid in making decisions. Aces andlOs favor you because naturals are worth half againmore to you than they are to the dealer. Small cards favorthe dealer by decreasing the dealer's chance of busting.

The high-low system, first introduced in 1963 byHarvey Dubner, is both simple and powerful. Edward O.Thorp in the revised edition of Beat the Dealer, LawrenceRevere in Playing Blackjack as a Business, and JulianBraun in How to Play Winning Blackjack, presentstrategy numbers for it. This chapter contains my independent calculations for it.

Peter Griffin in The Theory Of Blackjack says thebetting correlation of the high-low system is 0.97 andthe playing correlation is 0.51.

Counting cards in the high-low system is relativelysimple. Start with a count of zero after the cards areshufiled. Add one for every small card (2, 3, 4, 5, 6) that

Table 2High-Low Count

Card Count2,3,4,5,6 +1

7,8,9 01O,J,Q,K,Ace -1


you see. Subtract one for every ace or 10-count card thatyou see. Do not change your count for 7, 8, or 9. This isshown in table 2.

Keep a running count. Accumulate the total sincethe last shufile. A full deck contains the same number of+1 cards as -1 cards. Therefore, at the end of the deck therunning count should come back to zero. This is called a"balanced count."

Count Per DeckThis section explains count per deck, also called true

count.The high-low count tells you when the best cards,

the 1Os and aces, outnumber the small cards. The richerthe pack, that is, the more 1Os and aces relative to smallcards, the better for you. You need to know how rich thepack is in order to make appropriate bets and decisionsin play.

Richness of the pack depends on the proportion ofexcess 1Os and aces. For decision purposes, you mustrelate the running count to the number of decks youhave not seen. For example, twelve lOs and aces remaining to be used when the dealer is halfway through asingle deck is two 1Os and aces more than average; thatis as favorable to the player as if there were four excess1Os and aces in 52 cards, and is described as a count perdeck of +4.

For betting and playing decisions, a running countof +1 with half a deck remaining is equivalent to arunning count of +2 with one deck remaining, to arunning count of +4 with two decks remaining, and to arunning count of +8 with four decks remaining. Therefore, you must convert the running count into count per

HIGH-LOW SYSTEM 33

deck for making decisions. Simply divide the runningcount by the number of decks (or fraction of a deck) thatyou have not seen. If less than one deck remains, yourcount per deck will be greater than your running count.If more than one deck remains, your count per deck willbe less than your running count.

For example, suppose you approach a blackjacktable with a single-deck game in progress and see thefollowing cards.

First Player Second Player2-3-2-4-7 10-10

Dealer3-8-9

You glance at the cards and see that the runningcount is +3. Since what you have not seen is almost onedeck, the count per deck is slightly above +3. The dealerpicks up the cards, and you place a bet. You receive 2-10,the dealer's upcard is 2, the first other player has anatural, and you do not see the second other player'scards. You must decide whether to hit or to stand. Therunning count is now +2. Since about 2/3 of the deckremains unseen, you divide by 2/3. This is the same asmultiplying by 3/2. It gives a count per deck of +3. So youstand instead of hitting.

You need only approximate the count per deck. Youdo not need an exact count of the number of cardsremaining. If you had seen roughly half a deck in asingle-deck game, a running count of +2 would translateto a count per deck of about +4. If you had seen about halfa deck (26 cards) in a double-deck game, a running countof +2 would translate to a count per deck slightly greaterthan +1 (2 divided by 1.5 gives 4/3). A rough estimate ofthe count per deck suffices for decisions because you


need to know only whether it exceeds an integer in atable.

Estimating Your EdgeYour advantage or disadvantage if you play basic

strategy varies with the rules and number of decks used.Current Blackjack News is a good source for keeping upto date with the rules at various casinos and youradvantage or disadvantage. Commonly you are at adisadvantage of about 0.5% with basic strategy.

Table 3 contains more detail on estimating thecasino's edge right after a shuffle. The starting pointdepends on the number of decks used. Simulation showsthat a basic-strategy player playing with the benchmark rules has an expectation of 0.00% for one deck,-0.32% for two decks, and -0.5% for four or more decks.The negative signs mean the casino has an edge. Therest of table 3 contains numbers that you can add to oneof those starting points, depending on how the rules youface vary from those benchmark rules.

Early surrender gives the player 0.7%, of which0.4% is against an ace and 0.3% is against 10.

Table 3 allows you to find your expectation for thefirst hand after a shuffle. These are simulation results,and thus contain a small amount of random error. To asmall extent, the effect of a given rule also depends onthe number of decks used. The best presentation of howyour edge after a shuftle depends on the rules is in PeterGriffin's The Theory of Blackjack. You also can generatesuch numbers yourself with Blackjack Count Analyzerif you ask for a shuffle after every hand.

Table 3 applies to the first hand after a shuflle. Forsubsequent hands, your edge might be higher or lower.

HIGH-LOW SYSTEM

Table 3Player Edge

35

benchmark: one deck 0.00%benchmark: two decks -0.32benchmark: fOUf decks -0.47benchmark: six decks -0.52benchmark: eight decks -0.55

to the above add the following:rule change in player edge

dealer hits soft seventeen -0.20%double after splits O. 14double ten & eleven only -0.17double nine, ten & eleven only -0.08resplit aces (if four or more decks) 0.08late surrender 0.08early surrender 0.71lose all on doubles/splits versus natural -0.11

For each point increase in the count per deck, youradvantage goes up by 0.5%. (It goes up by increasingamounts at high counts, presumably due to insurancebecoming more valuable the higher the count.) Generally, in a multiple-deck game, when you have a count perdeck of +1 you are playing even with the casino - noadvantage or disadvantage. At a count per deck of +3 youhave an advantage of about 1%.

The 0.5% per count per deck works both ways: Thedealer's advantage over you increases with negativecounts. At a count per deck of -1 you are at a disadvantage of about 1% - if you play perfectly. If the count per


deck is -4, you can expect to lose at a rate of about 2.5%,which is 2% faster than if the dealer were to shuftle.

Winning with card counting requires betting moremoney when you have an edge than when you are at adisadvantage, and playing your hands correctly.

Jumping into a Game inProgress

When the best available table is a game in progressand you want to jump in without waiting for a shuflle,just start counting from zero and play according to yourcount. Treat the unseen discards the same as cards yetto be dealt. An unseen card is an unseen card whetherit has already been used or remains to be dealt. Example:You jump into a four-deck shoe after about one deck hasbeen used. You play and count as another deck is used.If your running count is +6, what is your count per deck?Two of the four decks have been used, but you havecounted the cards in only one deck. Since about threedecks remain unseen by you, divide your running countby three to get a count per deck of +2.

Effect of Rules on StrategySome casinos use one deck, some use six, and some

use other multiples. Strategy numbers vary only slightlywith the number of decks used. The tables in thischapter should be thought of as based on multiple decks.They actually are based on four decks, but are almostexactly the same as strategy numbers based on othermultiples such as two, six, etc. One-deck strategy numbers are slightly different from multiple-deck strategynumbers. You can use Blackjack Count Analyzer soft-

HIGH-LOW SYSTEM 37

ware to find strategy numbers tailored to any number ofdecks and any set of rules.

You should truncate, and not round, when usingthese tables. Examples: If the count per deck requiredto double down is +4, then double down only if youhave a count per deck of +4 or more, and do not doubledown if your count per deck is say 3.8. If a count perdeck of -1 is required to stand, then stand if your countper deck, truncated, is -lor higher. (Zero is higher than-1, and a count per deck of -1.8 is truncated to -1.)

At some casinos dealers stand on soft seventeen,whereas at other casinos dealers hit it. The discussionof this chapter initially assumes that the dealer standson soft seventeen, and then also covers what to dowhen the dealer hits soft seventeen.

The decisions are discussed in the order in whichyou make them at a casino: insurance, pair splits,double down, and hit or stand. The strategy numbersin this book are derived for multiple decks, and are alsovery close for one deck. You must memorize them; whenyou are playing blackjack in a casino, you can refer tothese strategy numbers only in your head.

InsuranceAdvice of well-meaning but ill-informed gamblers

that you should insure only a natural is worth its cost- nothing; you should buy insurance if more than onethird of the unseen cards are lOs. As soon as you see thedealer's ace, begin considering whether you should buyinsurance. If the count per deck is greater than orequal to +3, buy insurance.


PairsTable 4 covers pair splits, assuming that you cannot

double down after splitting. If you can double down aftersplitting, then use table 5 instead.

Count per deck also determines whether you shouldsplit pairs. The higher the count per deck, the more pairsyou should split. The exception is 8-8 against 10, whichyou should split when the count per deck is less than +6and not split when the count per deck is +6 or more.

Table 4Split, With No Double After,

S17Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 Aace-ace spl spl spl spl spl -9 -8 -7 -8 -310-10 8 6 5 49-9 0 -2 -3 -4 -4 6 -8 -9 38-8 spl spl spl spl spl spl spl spl 6* spl7-7 -9 spl spl spl spl spl6-6 2 0 -3 -5 -75-54-43-3 8 3 0 -2 -9 spl2-2 7 3 0 -4 -9 spl

KEYblank Do not split.

spl Split.

s17 Dealer stands on soft seventeen.

number Split if the count per deck equals or exceeds thenumber; do not split if the count per deck is less thanthe number.

* Reverse the meaning. Split only if the count per deckis less than the number in the table.

HIGH-LOW SYSTEM 39

If you do not split, then play your hand according tothe strategy number for the total.

Examples:A) You get 3-3 against 3. The number from table

4 is +3. If the count per deck equals or exceeds +3,you should split the pair. If the count per deck isless than +3, you should hit.

B) You get 3-3 against 9. That spot in table 4 isblank, so do not split, no matter what the count perdeck.

C) You get 8-8 against 9; table 4 says to split nomatter what the count per deck.

Table 5Split, With Double After, S17

Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 Aace-ace spl spl spl spl spl -9 -8 -7 -8 -310-10 8 6 5 49-9 -2 -3 -5 -6 -6 3 -8 -9 38-8 spl spl spl spl spl spl spl spl 8* spl7-7 -9 spl spl spl spl spl 56-6 -1 -4 -6 -8 -105-54-4 6 1 -1 -43-3 0 -4 -7 -9 spl spl 42-2 -3 -5 -7 -9 spl spl 5

KEY

blank Do not split.

spl Split.





Doubling DownDoubling down also depends on count per deck. You

should be more likely to double down 'if the count perdeck is higher. If you decide to double down, always putup additional money equal to the full amount of youroriginal bet.

The soft-doubling parts of all double-down tables inthis book assume that the ace in your hand can be valuedat either 1 or 11 after you split. If the ace cannot swing,then do not double down on soft hands of nineteen orless.

Examples:A) You get 6-5 against 4. Your hand totals eleven,

and table 6 says to double down no matter what thecount per deck.

B) You get 4-5 against 9. Your hand totals nineand table 6 says do not double down, so you take ahit.

e) You get ace-8 against 6. You may call yourtotal nine and double down if you wish, and table6 advises doing so if the count per deck equals orexceeds +1. If the count per deck is less than +1, orif doubling and catching a 2 would give you eleveninstead of twenty-one, call your total nineteen andstand.

HIGH-LOW SYSTEM 41

Player'sHand111098**7

ace-9ace-8ace-7ace-6ace-5ace-4ace-3ace-2

KEY

blank

db

s17

number

**

Table 6Double Down, S17

Dealer's Upcard2 3 4 5 6 7 8 9 10 Adb db db db db -9 -6 -4 -4 1-8 -9 -10 db db -6 -4 -1 4 41 0 -2 -4 -6 3 7

9 5 3 19 9

10 8 6 5 48 5 3 1 10 -2 -6 -8 -101 -3 -7 -10 db

4 -2 -6 db7 0 -4 -97 1 -1 -47 3 0 -1

Do not double down.

Double down.


Double down if the count per deck equals or exceedsthe number; do not double down if the count per deckis less than the number.

With 4-4, splitting is better than doubling down ifdoubling is allowed after splitting.


Soft StandingTable 7 summarizes soft standing decisions. Soft

doubling as in table 6 takes precedence, so use table 7only if you are not doubling down.

If you do not double down you should hit a soft totalof seventeen or less and stand with a soft total ofnineteen or more. With soft eighteen if you do not doubledown you should stand against 2 through 8, and hitagainst 9 or 10. Against ace you should hit soft eighteenwhen the count per deck is less than +1, but stand whenthe count per deck is +1 or more.

Table 7Soft Standing, S17

Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 Asoft 19soft 18 h h 1soft 17 h h h h h h h h h h

KEY

blank Stand.

h Hit.

s 17 Dealer stands on soft seventeen.

number Stand if the count per deck exceeds the number; hit ifthe count per deck is less than the number; flip a coinif the count per deck equals the number.

HIGH-LOW SYSTEM 43

Hard StandingTable 8 covers hard standing. With a total of eleven

or less, never stand. With a hard total of twelve or more,you should be more likely to stand when the count perdeck is higher.

Examples:A) You have 9-8 for a total of seventeen against

2; table 8 says to stand.B) You have 10-4 for fourteen against 8; table 8

says to hit.e) You have 2-6 for a total of eight against 10;

you ask for a hit. Suppose you receive a 7 to bringyour total to fifteen. If the count per deck is lessthan +4, you should ask for another hit; if the countper deck equals or exceeds +4, you should stand.

Table 8Hard Standing, S17

Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 Ahard 17 -6hard 16 -8 -10 9 7 5 0 8hard 15 -5 -6 -7 -9 -9 10 10 8 4 10hard 14 -3 -4 -6 -7 -7 h h h h hhard 13 0 -1 -3 -4 -4 h h h h hhard 12 3 2 0 -1 0 h h h h h

KEY

blank Stand.

h Hit.

s17 Dealer stands on soft seventeen.number Stand if the count per deck equals or exceeds the

number; hit if the count per deck is less than thenumber.


If the Dealer Hits Soft SeventeenTables 4-8 assume that the dealer stands on soft

seventeen. If you are playing blackjack at a casino wherethe dealer hits soft seventeen, you should play slightlydifferently, primarily against an ace. The differencesagainst an ace are: You will be more likely to hit softeighteen, more likely to stand on hard sixteen or hardfifteen, more likely to double down on eleven, and morelikely to split 9-9. Hitting soft seventeen has theseeffects on strategy because the dealer is less likely tofinish the hand with exactly seventeen and is morelikely to bust.

The major changes to make to tables 4-8 if the dealerhits soft seventeen are as follows:

Table 7: Always hit soft 18 against ace.

Table 8: With hard 16 against ace, stand if the countper deck is +3 or more.

With hard 15 against ace, stand if the countper deck is +5 or more.

With hard 12 against 6, stand if the countper deck is -3 or more.

Simulation of the dealer hitting soft seventeenresults in a win mte lower than the benchmark by $5 per100 hands. So the dealer hitting soft seventeen is a rulevariation that is costly to the card counter.

BenchmarkWin rate: $16Std dev: $415

Dlr hits soft 17Win rate: $11Std dev: $416

HIGH-LOW SYSTEM

Summary Strategy Tables

45

Tables 9 and 10 summarize strategy numbers forthe high-low count. Table 9 applies if the dealer standson soft seventeen, and table 10 applies if the dealer hitssoft seventeen. Table 10 includes the major changeslisted above plus minor changes.

Soft eighteen is represented by two lines; the -/dbline is for deciding whether to stand or double down, andthe h/- line is for choosing between hit and stand. Theline for doubling takes precedence. Note that in table 9,the 1 for ace-7 against ace is for choosing between hitand stand, and does not say to double down.

Importance of Strategy TablesThis chapter has presented multiple-deck playing

strategy, but at some casinos blackjack is dealt with onedeck. Multiple-deck strategy numbers are only slightlydifferent from single-deck strategy numbers. You willwin almost as much by using multiple-deck strategynumbers with one deck as you will win by using onedeck strategy numbers with one deck. With freshlyshuftled cards, the casino has a larger advantage as thenumber of decks increases.

You might be wondering how important bet variation is relative to strategy variation.

Using basic strategy and flat $26.54 bets, sittingthrough shoe after shoe, loses at the rate of $15 per hour.So the total improvement from using the high-lowsystem is from -$15 to +$16, or $31.

Playing hands according to' basic strategy and usingthe high-low for bet variation wins at the rate of $12 perhour. So bet variation alone improves on flat-betting by

(text continues on page 50)


Table 9Strategy Numbers, S17


double not allowed after splitA-A spl spl spl spl spl -9 -8 -7 -8 -310-10 8 6 5 49-9 0 -2 -3 -4 -4 6 -8 -9 38-8 spl spl spl spl spl spl spl spl 6* spl7-7 -9 spl spl spl spl spl h h h h6-6 2 0 -3 -5 -7 h h h h h5-5 never split - play as total of ten4-4 never split - play as total of eight3-3 8 3 0 -2 -9 spl h h h h2-2 7 3 0 -4 -9 spl h h h h

ace-9 10 8 6 5 4ace-8 8 5 3 1 1ace-7 -/db 0 -2 -6 -8 -10 h h h/-ace-7 hI- h h 1ace-6 1 -3 -7 -10 db h h h h hace-5 h 4 -2 -6 db h h h h hace-4 h 7 0 -4 -9 h h h h hace-3 h 7 1 -1 -4 h h h h hace-2 h 7 3 0 -1 h h h h h

hard 20hard 19hard 18hard 17 -6hard 16 -8 -10 9 7 5 0 8hard 15 -5 -6 -7 -9 -9 10 10 8 4 10hard 14 -3 -4 -6 -7 -7 h h h h hhard 13 0 -1 -3 -4 -4 h h h h hhard 12 3 2 0 -1 0 h h h h h

11 db db db db db -9 -6 -4 -4 110 -8 -9 -10 db db -6 -4 -1 4 49 1 0 -2 -4 -6 3 7 h h h8** h 9 5 3 1 h h h h h7 h h h 9 9 h h h h h6 h h h h h h h h h h5 h h h h h h h h h h

HIGH-LOW SYSTEM 47


double allowed after splitA-A spl spl spl spl spl -9 -8 -7 -8 -310-10 8 6 5 49-9 -2 -3 -5 -6 -6 3 -8 -9 38-8 spl spl spl spl spl spl spl spl 8* spl7-7 -9 spl spl spl spl spl 5 h h h6-6 -1 -4 -6 -8 -10 h h h h h5-5 never split - play as total of ten4-4** h 6 1 -1 -4 h h h h h3-3 0 -4 -7 -9 spl spl 4 h h h2-2 -3 -5 -7 -9 spl spl 5 h h h

KEY

-/db

db

hh/-

spl

s17

number

*

**

Stand.

Do not hit. See the -/db row for the number sayingwhether to stand or double down.

Double down.

Hit.

Do not double. See the h/- row for the number sayingwhether to hit or to stand.

Split.


Stand (or double or split) at a count per deck equal toor greater than the number; hit (or do not split) at acount per deck less than the number. (Note that the1 for ace-7 against ace is hit/stand, and does not sayto double down.)

Reverse the meaning. Split only if the count per deckis less than the number in the table.



Table 10Strategy Numbers, HI7


double not ai/owed after splitA-A spl spl spl spl spl -9 -8 -7 -8 -410-10 8 6 5 49-9 -1 -2 -3 -4 -6 6 -8 -9 28-8 spl spl spl spl spl spl spl spl 6* -17-7 -8 -10 spl spl spl spl h h h h6-6 1 -1 -3 -5 -7 h h h h h5-5 never split - play as total of ten4-4 never split - play as total of eight3-3 8 3 0 -2 -5 spl h h h h2-2 7 3 0 -4 -7 spl h h h h

ace-9 10 8 6 5 4ace-8 8 5 3 1 0ace-7 -/db 0 -2 -6 -8 db h h hace-7 hI- h h hace-6 1 -3 -7 -10 db h h h h hace-5 h 3 -3 -6 db h h h h hace-4 h 6 0 -4 -10 h h h h hace-3 h 6 1 -1 -5 h h h h hace-2 h 7 3 0 -2 h h h h h

hard 20hard 19hard 18hard 17 -4hard 16 -9 -10 9 7 5 0 3hard 15 -5 -7 -8 -9 10 10 8 4 5hard 14 -3 -5 -6 -7 -9 h h h h 9hard 13 0 -2 -3 -4 -7 h h h h hhard 12 3 1 0 -1 -3 h h h h h] ] db db db db db -9 -6 -4 -4 010 -8 -9 -10 db db -6 -4 -1 4 39 1 0 -2 -4 -6 3 7 h h h8** h 9 5 3 1 h h h h h7 h h h 9 9 h h h h h6 h h h h h h h h h h5 h h h h h h h h h h

HIGH-LOW SYSTEM 49


double allowed after splitA-A spl spl spl spl spl -9 -8 -7 -8 -410-10 8 6 5 49-9 -2 -4 -5 -6 -8 3 -8 -9 18-8 spl spl spl spl spl spl spl spl 8* -17-7 -10 spl spl spl spl spl 5 h h h6-6 -2 -4 -6 -8 -10 h h h h h5-5 never split - play as total of ten4-4** h 6 1 -1 -6 h h h h h3-3 0 -3 -7 -9 spl spl 4 h h h2-2 -2 -5 -7 -9 spl spl 5 h h h

KEY

db

hh17

h/-

spl

number

*

**

Stand.

Double down.

Hit.

Dealer hits soft seventeen.

Do not double. See the hI- row for the number sayingwhether to hit or to stand.

Split.

Stand (or double or split) at a count per deck equal toor greater than the number; hit (or do not split) at acount per deck less than the number.




$27 per hour, or 87% of the $31 gain from both betvariation and strategy variation.

Playing only positive counts and leaving the tableon negative counts, as popularized by earlier editions ofProfessional Blackjack, is a form of bet variation.

Using flat $26.54 bets hand after hand but varyingstrategy according to the count loses at the rate of $14.34per hour. That number comes from the same 600million-hand simulation that gave the benchmark winrate; the standard error on it is $0.14. In the box below,the -$14.34 is rounded off to -$14. So varying strategywith the count per deck improves on basic strategy byonly a small amount when flat bets are used in a six-deckgame. Most of that improvement is due to taking insurance, standing on hard sixteen against 10, and standingon hard fifteen against 10, all at the appropriate timesof course. After those three decisions, the amounts a cardcounter gains by correctly deviating from basic strategydrop off dramatically. See Donald Schlesinger's "Attacking the Shoe" in the September 1986 issue of BlackjackForum for more detail on this point.

Flat $26.54 bets,basic strategy

Win rate: -$15Std dev: $301

Flat $26.54 bets, varystrategy with count

Win rate: -$14Std dev: $300

Vary bets with count,basic strategy

Win rate: $12Std dev: $401

HIGH-LOW SYSTEM 51

As a practical matter, you do not really need a widerange of strategy numbers. I simulated the benchmarkmodified to use only strategy numbers -1 to +6, and gota win rate of $1 7 per hour. Using a smaller range ofstrategy numbers should result in a lower win rate, butI got a higher rate. I increased the sample sizes to seewhether the gap would disappear or the differencewould become statistically significant. It became significant. This is a puzzle, and I am continuing to work on it.My suggestion for beginners, and for anyone else whowants to cut down on the volume of numbers to memorize, is ignore all strategy numbers smaller than -1 andlarger than +6.

For winning money at multiple-deck blackjack, betvariation is much more important than strategy variation. In fact, all of these tables on deviating from basicstrategy are of minor value. More important is to avoidbeing barred and to find good rules and good penetration.

Appendix A contains more details on strategy rulesfor the high-low system.


CHAPTER 4

INSURANCE

Insurance is a side bet that the dealer has a natural.Table 11 gives the minimum counts per deck for whichinsurance is a good buy, as a function of number of decksshuffled together. For example, if six decks are used,insurance is likely to be profitable if the count per deck,after counting the dealer's ace and as many other cardsas you can see, exceeds 3.0; that 3.0 applies to the sixdeck game no matter how many decks are yet unseen.

My suggestion to simplify the strategy numbers:Use 1.4 for one deck, 2.4 for two decks, and 3 for morethan two decks.

No insuranceWin rate: $14Std dev: $416

INSURANCE

Table 11VVhen to Take Insurance

53

Decks High-Low1 1.42 2.43 2.74 2.95 3.06 3.07 3.18 3.1

infinite 3.3

Halves1.72.73.03.23.33.33.43.43.7

Peter Griffin and Arnold Snyder assisted with themathematics involved in calculations for table 11. Table11 also includes advice for the halves counting system,which is explained in chapter 12. Take insurance at acount per deck equal to or greater than the strategynumber shown in table 11.

The insurance number does not depend on whetheror not the dealer takes a hole card.

For six decks, ignoring insurance wins about $2 per100 hands less than the benchmark.

For single-deck games, insurance is quite valuable.The following boxes report the results of two simulations, one with and one without insurance. The shufllepoint is 26 cards yet unseen.

One deck, ins.Win rate: $48Std dev: $482

One deck, no ins.Win rate: $41Std dev: $484


A 10-Count for InsuranceIt is possible to use a IO-count to make better

insurance decisions. The IO-count is the best count forinsurance. The only way to make better insurancedecisions is to get a glimpse of the dealer's hole card.Using a IO-count for insurance in addition to the highlow for betting and playing the hands adds about $2 perhour to the hourly win rate for six decks. At single deck,using a IO-count in addition to the high-low adds about$8 per hour to the hourly win rate.

The simplest way to do the 10-count is to keep thefollowing as a second count as shown in table 12-: assignthe value +1 to all cards except 10 and assign the value-2 to every 10. If this running IO-count gets above fourtimes the number of decks shuftled together (+4 with asingle deck, +8 with a double deck, +16 with four decks,or +24 with six decks), buying insurance is profitable.Note that the 10-count insurance decision is based onthe running count and not on the count per deck. Forexample, to buy insurance on a six-deck game you needa IO-count of +24 or more no matter how many deckshave yet to be dealt. Also note that the pluses andminuses do not balance out - when all the cards aregone, your running count will be four times the numberof decks used.

Table 12to-Count for Insurance

Cardace, 2, 3, 4, 5, 6, 7, 8, 9IO,J,Q,K

Count+1-2

INSURANCE 55

Suppose the dealer shuflles a single deck and dealsthe following cards:

ace-9 6-6-8 7-10After that round the high-low running count is zero.

Suppose you place a bet, are dealt 8-9, and the dealershows an ace. Would you buy insurance? Though thehigh-low count of -1 says no, you should. The reason isthat an excess of 7s, 8s, 9s, and aces has appeared. Of the42 cards you have not seen, 15 are lOs; when you buyinsurance you will be paid 2 to 1 for something that willhappen 15 out of 42 times, giving you an expected-winrate of roughly 7%. If you are keeping a IO-count forinsurance decisions, the IO-count would be +7 and thusindicate that you should buy insurance.

A single running count that relates 1Os to non-lOsfor accuracy on insurance could be used for all decisions,but such a count is less accurate for estimating advantage; gains resulting from accuracy on insurance aremore than offset by losses on accuracy on estimatingadvantage. A simulation with a IO-count for all decisions shows that it falls short of the benchmark by $8 perhour. For single deck, use of a IO-count for all decisionsyields results short of the single-deck benchmark by $19per hour! IO-count strategies are obsolete for this reason.

6 decks, add 10-countfor insurance


1 deck, add lO-countfor insurance



6 decks, 10-count only 1 deck, 10-count onlyWin rate: $7 Win rate: $29Std dev: $336 Std dev: $412

Do not use table 11 if you are keeping a lO-count,because the lO-count gives positively accurate insurance guidance.

Bustout BetSome casinos offer a bustout bet. Winning bustout

bets pay 2:1. If the dealer's first two cards total twelvethrough sixteen and you still have a hand in play, youcan bet that the next card will be a 10 or face card. If youbust, or if you have a natural for which you have alreadybeen paid, you do not have the option of making thebustout bet. If you split or double down, the amount youcan bet on the bustout is not doubled.

If you use the high-low or halves (see chapter 12)counting system with the bustout bet, use 3.33 as thedecision number. The reason the insurance number doesnot apply to the bustout bet is the dealer does not showan ace.

With basic strategy, you will have the option ofmaking the bustout bet 32% of the time, or four times asoften as the dealer shows an ace. The max on the bustoutbet is your original bet. So the bustout bet is approximately eight times as profitable as insurance. Withinsurance the dealer always shows an ace, and insurance is a better bet the more non-lOs have been removed

INSURANCE 57

from the pack. But with the bustout bet, you get to seemore cards before deciding whether to make the bet.

My suggestion is to play bust-out blackjack with apartner, keeping two counts and helping each othermake decisions. One of you uses a ten-count to makecorrect bust-out and insurance bets, and the other usesa conventional counting system (such as high-low) forbetting and making decisions other than bustout andinsurance.

Bustout, six decksWin rate: $30Std dey: $517

Bustout, one deckWin rate: $95Std dey: $606

One deck, add 10-countfor bustout & insurance

Win rate: $107Std dey: $583

Six decks, add 10-countfor bustout & insurance



CHAPTER 5

DOUBLE DOWN

Doubling down when the dealer stands on softseventeen was presented as table 6 (page 41), and isreproduced here for your convenience as table 13. Table14 is doubling down when the dealer hits soft seventeen.As you can see by comparing those two tables, how thedealer plays soft seventeen has almost no effect onwhether you should double down.

If you can double down on certain totals only, thenuse only the appropriate part of the tables. For example,if you can double down on ten or eleven only, then ignorethe parts of the tables that give advice on doubling downon nine or less.

Double on Any Number of CardsTables 13 and 14 are for use when this is your only

chance to double, and if you hit instead you no longer

DOUBLE DOWN 59

have the option of doubling. This is the most commonform of doubling down.

At a few casinos, you can double down on anynumber of cards. This rule makes doubling on small twocard hands less attractive. The strategy numbers fordoubling down when you can double on any number ofcards are shown in tables 15 and 16. With ace-2 or ace3 when the count is high enough, the tables say to doubledown against 4, 5, or 6; however, you might as well justhit all hands of A-2 and A-3 because the value ofdoubling is approximately equal to the value of hittingno matter how high the count. That is true no matterwhich action the dealer takes on soft seventeen.

Double on Two or Three CardsAt Las Vegas Club in Las Vegas, starting in 1993,

you can double down on two or three cards, except aftersplitting. This requires two double-down strategies. Onestrategy (table 16) is for use if you will still have theoption of doubling down after hitting once, and the otherstrategy (table 14) is for use if this is your last chance todouble on this hand. For example, with eight against 5and a high count, you ought to hit if you will have theoption of doubling after that hit, but double down now ifyou will not have the option of doubling after hitting.

With Multiple-Card SurrenderIn some Asian casinos you can (or could) surrender

on any number of cards. This makes hitting moreattractive relative to doubling down. The appropriatedoubling strategy is shown in table 17. The only changefrom table 13 is the strategy number for ten against 10is +6.


60

Player'sHand111098**7


KEY

blank

db

s17

number

**

PROFESSIONAL BLACKJACK

Table 13Double Down, S17

Dealer's Upcard2 3 4 5 6 7 8 9 10 A

db db db db db -9 -6 -4 -4 1-8 -9 -10 db db -6 -4 -1 4 41 0 -2 -4 -6 3 7

9 5 3 19 9

10 8 6 5 48 5 3 1 10 -2 -6 -8 -101 -3 -7 -10 db

4 -2 -6 db7 0 -4 -97 1 -1 -47 3 0 -1

Do not double down.

Double down.




Double 10 & 11Win rate: $10Std dey: $400

Double 9, 10 & 11Win rate: $12Std dey: $406

DOUBLE DOWN 61

Player'sHand

111098**7


KEY

blank

db

h17

number

**

Table 14Double Down, HI7


db db db db db -9 -6 -4 -4 0-8 -9 -10 db db -6 -4 -1 4 31 0 -2 -4 -6 3 7

9 5 3 19 8

10 8 6 5 48 5 3 1 00 -2 -6 -8 db1 -3 -7 -10 db

3 -3 -6 db6 0 -4 -106 1 -1 -57 3 0 -2

Do not double down.

Double down.





Table 15Double, Not Last Chance, S17

Player'sHand

111098**7

soft 21soft 20soft 19soft 18soft 17soft 16soft 15soft 14soft 13

KEY

blank

db

s17

number

**


db db db db db -9 -6 -4 -4 1-8 -9 -10 db db -6 -4 -1 4 42 0 -1 -2 -4 4 8

9 8

10 8 6 5 48 5 3 1 10 -2 -6 -8 -101 -3 -7 -10 db

0 -5 db8 -1 -68 3 09 4 7

Do not double down.

Double down.



Double down on 4-4 rather than split only if doublingdown is not permitted after pair splitting.

DOUBLE DOWN 63


db db db db db -9 -6 -4 -4 0-8 -9 -10 db db -6 -4 -1 4 32 0 -1 -2 -4 4 8

9 8

Table 16Double, Not Last Chance, Ht7

Player'sHand

111098**7

soft 21soft 20soft 19soft 18soft 17soft 16soft 15soft 14soft 13

KEY

blank

db

h17

number

**

10 8 6 5 48 5 3 1 00 -2 -6 -8 db1 -3 -7 -10 db

0 -5 db5 -1 -78 2 -19 4 2

Do not double down.

Double down.



Double down on 4-4 rather than split only if doublingdown is not permitted after pair splitting.


Table 17Double Down, With

Multiple-Card Surrender, S17Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 A

11 db db db db db -9 -6 -4 -4 110 -8 -9 -10 db db -6 -4 -1 6 49 1 0 -2 -4 -6 3 78** 9 5 3 17 9 9

ace-9 10 8 6 5 4ace-8 8 5 3 1 1ace-7 0 -2 -6 -8 -10ace-6 1 -3 -7 -10 dbace-5 4 -2 -6 dbace-4 7 0 -4 -9ace-3 7 1 -1 -4ace-2 7 3 0 -1

KEY

blank Do not double down.

db Double down.

s!7 Dealer stands on soft seventeen.

number Double down if the count per deck equals or exceedsthe number; do not double down if the count per deckis less than the number.

** With 4-4, splitting is better than doubling down ifdoubling is allowed after splitting.

DOUBLE DOWN 65

Table 18Double Down, With

Surrender After Doubling, S17Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 A

11 db db db db db -9 -7 -6 -5 -510 -8 -9 -10 db db -6 -4 -2 0 -19 1 0 -2 -4 -6 3 68** 9 5 3 17 9 9

ace-9 10 8 6 5 4ace-8 8 5 3 1 1ace-7 0 -2 -6 -8 -10ace-6 1 -3 -7 -10 dbace-5 4 -2 -6 dbace-4 7 0 -4 -9ace-3 7 1 -1 -4ace-2 7 3 0 -1

KEY

blank Do not double down.

db Double down.


number Double down if the count per deck equals or exceedsthe number; do not double down if the count per deckis less than the number.

** With 4-4, splitting is better than doubling down ifdoubling is allowed after splitting.


For dealers hit soft seventeen, use the ace column oftable 14 with the rest of table 17.

With Surrender After DoublingIf you have multiple-card surrender plus surrender

after doubling down, use the strategy numbers fromtable 18. Surrender strategy can be found in the surrender chapter.

If the dealer hits instead of stands on soft seventeen,the only change to the ace column of table 18 is thestrategy number for doubling down on ten against ace;it should be o.

67

CHAPTER 6

SPLIT

Table 4 (page 38) covers pair splits, assuming thatthe dealer stands on soft seventeen and you cannotdouble down after splitting; it is reproduced here astable 19. Table 5 (page 39) covers pair splits if the dealerstands on soft seventeen and you can double down aftersplitting; it is reproduced here as table 20.

Tables 21 and 22 are parallel, and apply if the dealerhits soft seventeen. Table 21 contains the strategynumbers for splits if the dealer hits soft seventeen andyou cannot double down after splitting. Table 22 contains the strategy numbers for splits if the dealer hitssoft seventeen and you can double down after splitting.

Note that the asterisk reverses the meaning of thestrategy number. Normally you split if the count perdeck equals or exceeds the number. But if there is an


asterisk, you split if the count per deck is less than thenumber.

Comparison of the four tables with each other showsthat being able to double down on any two cards aftersplitting has large effects on playing strategy, but whetherthe dealer hits or stands on soft seventeen makes adifference only for splitting 8-8 against ace.

Double after splits is a valuable option, adding $4per hour to the benchmark win rate.

ResplitsResplits make no difference in the sense that the

strategy for splits is the same whether or not resplits areallowed.

Resplits follow split strategy: The strategy numberfor resplits is the same as the strategy number for splits.

Not being able to resplit costs you about a dollar per100 hands.

Double Any Number of CardsSuppose you can double down on any number of

cards. This rule makes splitting small pairs such as 3-3less attractive, because you might hit the total (6) witha small card, such as a 4 or 5, bringing your hand to atotal that would be an attractive three-card double.Tables 23 and 24 contain strategy numbers for splittingif you can double down on any number of cards, butcannot double down after splitting. Table 23 is for dealerstands on soft seventeen; and table 24 is for dealer hitssoft seventeen.

Being able to double down on any number of cardsincreases the value of hitting small totals, including

SPLIT 69

pairs. But being able to double down on any number ofcards after splitting increases the value of splitting. Thestronger effect is the increased value of being able todouble down after a hit. Therefore if you can doubledown on any number of cards, you are less likely to splitsmall pairs.

Tables 25 and 26 contain strategy numbers forsplitting if you can double down on any number of cardsafter splitting; table 25 is for dealer stands on softseventeen; and table 26 is for dealer hits soft seventeen.

In 1993 Las Vegas Club in Las Vegas introduced ahybrid: you can double down on any two or three cardsbefore you split, but you can double down only on anytwo cards after you split. You cannot double down on fouror more cards, and after splitting you cannot doubledown on three cards. Tables 27 and 28 cover Las VegasClub splitting. Table 27 is for dealer stands on softseventeen, and table 28 is for dealer hits soft seventeen.


No resplitsWin rate: $15Std dev: $413



S17Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 Aace-ace spl spl spl spl spl -9 -8 -7 -8 -310-10 8 6 5 49-9 0 -2 -3 -4 -4 6 -8 -9 38-8 spl spl spl spl spl spl spl spl 6* spl7-7 -9 spl spl spl spl spl6-6 2 0 -3 -5 -75-54-43-3 8 3 0 -2 -9 spl2-2 7 3 0 -4 -9 spl

KEY

blank Do not split.

spl Split.




SPLIT

Table 20Split, With Double After, S17

71

Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 Aace-ace spl spl spl spl spl -9 -8 -7 -8 -310-10 8 6 5 49-9 -2 -3 -5 -6 -6 3 -8 -9 38-8 spl spl spl spl spl spl spl spl 8* spl7-7 -9 spl spl spl spl spl 56-6 -1 -4 -6 -8 -105-54-4 6 1 -1 -43-3 0 -4 -7 -9 spl spl 42-2 -3 -5 -7 -9 spl spl 5

KEYblank Do not split.

spl Split.




Double after splitWin rate: $20Std dev: $421



HI7Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 Aace-ace spl spl spl spl spl -9 -8 -7 -8 -410-10 8 6 5 49-9 -1 -2 -3 -4 -6 6 -8 -9 28-8 spl spl spl spl spl spl spl spl 6* -17-7 -8 -10 spl spl spl spl6-6 1 -1 -3 -5 -75-54-43-3 8 3 0 -2 -5 spl2-2 7 3 0 -4 -7 spl

KEY

blank Do not split.

h17 Dealer hits soft seventeen.

spl Split.



SPLIT 73

Table 22Split, With Double After, Ht7

Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 Aace-ace spl spl spl spl spl -9 -8 -7 -8 -410-10 8 6 5 49-9 -2 -4 -5 -6 -8 3 -8 -9 18-8 spl spl spl spl spl spl spl spl 8* -17-7 -10 spl spl spl spl spl 56-6 -2 -4 -6 -8 -105-54-4 6 1 -1 -63-3 0 -3 -7 -9 spl spl 42-2 -2 -5 -7 -9 spl spl 5

KEY

blank Do not split.


spl Split.




Table 23Split, With Double Any Numberof Cards and No Double After

Split, S17Player'sHandace-ace10-109-98-87-76-65-54-43-32-2

KEY

blank

spl

s17

number

*


spl spl spl spl spl -9 -8 -7 -8 -38 6 5 4

0 -2 -3 -4 -4 6 -8 -9 3spl spl spl spl spl spl spl spl 6* spl-9 spl spl spl spl spl2 0 -3 -5 -7

9 6 2 2 28 4 1 0 5

Do not split.

Split.


Split if the count per deck equals or exceeds thenumber; do not split if the count per deck is less thanthe number.


SPLIT 75

Table 24Split, With Double Any Numberof Cards and No Double After

Split, HI7Player'sHandace-ace10-109-98-87-76-65-54-43-32-2

KEYblank

hl7

spl

number

*


spl spl spl spl spl -9 -8 -7 -8 -48 6 5 4

-1 -2 -3 -4 -6 6 -8 -9 2spl spl spl spl spl spl spl spl 6* -1-8 -10 spl spl spl splI -I -3 -5 -7

9 5 2 0 28 4 I 0 5

Do not split.


Split.

Split if the count per deck equals or exceeds thenumber; do not split if the count per deck is less thanthe number.



Table 25Split, With Double Any Numberof Cards Including After Split,

S17Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 Aace-ace spl spl spl spl spl -9 -8 -7 -8 -310-10 8 6 5 49-9 -2 -3 -5 -6 -6 3 -8 -9 38-8 spl spl spl spl spl spl spl spl 8* spl7-7 -9 spl spl spl spl spl 46-6 -1 -4 -6 -8 -105-54-4 9 3 63-3 3 0 -3 -5 spl spl2-2 0 -3 -6 -8 spl spl

KEY

blank Do not split.

spl Split.




Double any number of cards,including after splitting


SPLIT 77

Table 26Split, With Double Any Numberof Cards Including After Split,

HI7Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 Aace-ace spl spl spl spl spl -9 -8 -7 -8 -410-10 8 6 5 49-9 -2 -4 -5 -6 -8 3 -8 -9 18-8 spl spl spl spl spl spl spl spl 8* -17-7 -10 spl spl spl spl spl 46-6 -2 -4 -6 -8 -105-54-4 8 3 03-3 3 0 -3 -5 -8 spl2-2 0 -3 -6 -8 -10 spl

KEY

blank Do not split.


spl Split.




Table 27Split, With Double Any Two orThree Cards Before Split and

Any Two After, S17Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 Aace-ace spl spl spl spl spl -9 -8 -7 -8 -310-10 8 6 5 49-9 -2 -3 -5 -6 -6 3 -8 -9 38-8 spl spl spl spl spl spl spl spl 8* spl7-7 -9 spl spl spl spl spl 46-6 -1 -3 -5 -7 -105-54-4 9 3 63-3 3 0 -2 -4 spl spl2-2 2 -1 -5 -7 spl spl

KEY

blank Do not split.

spl Split.




Double three cards beforesplitting, two after


SPLIT 79

Table 28Split, With Double Any Two orThree Cards Before Split and

Any Two After, H17Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 Aace-ace spl spl spl spl spl -9 -8 -7 -8 -410-10 8 6 5 49-9 -2 -4 -5 -6 -8 3 -8 -9 18-8 spl spl spl spl spl spl spl spl 8* -17-7 -10 spl spl spl spl spl 46-6 -1 -3 -6 -7 -105-54-4 8 3 03-3 3 0 -2 -4 -6 spl2-2 2 -1 -4 -7 -10 spl

KEY

blank Do not split.


spl Split.




CHAPTER 7

NO HOLE CARD

At some casinos the dealer takes only an upcard,and no hole card. The players get two cards each to start,and finish their hands in the normal mamier. Then thedealer takes a second card, and if necessary, more cards.

Suppose the dealer gets a natural, a two-card twentyone. At most casinos you lose unless you also have anatural; if you have gotten to twenty-one in three ormore cards you lose if the dealer gets a natural.

At some casinos you lose only your original bet andnot any extra amounts you have wagered on doubles orsplits. At other casinos if you have doubled down or splitand the dealer gets a natural you lose your wholedoubled bet.

There also are casinos that allow you to keep yourmoney (i.e. not lose) if you hit to twenty-one in any

No HOLE CARD 81

number of cards and then the dealer's ace or 10 turnsinto a two-card twenty-one.

Lose Only Original Bet to aNatural

Whether the dealer's second card is face down underthe upcard or is in the pack yet to be dealt should notmatter to you. This is true even if you are the last playerto act. If you not taking a card means it will be thedealer's second card, the proper play of your hand is thesame as if the dealer already had a second card. If youhave sixteen and the proper play is to hit, then youshould hit whether the dealer has a hole card or not.

What does make a difference in strategy is whathappens when the dealer gets a natural.

Lose All to a NaturalIn Europe and a few other places, the dealer does not

take a hole card, and if the dealer ends up with a naturalyou lose your whole bet if you have doubled down or splita pair. This means you should be less willing to wagerextra money on your hand when the dealer shows a 10or ace.

If you cannot lose more than your original bet in theevent of a natural by the dealer, this section does notapply. This section applies only for the situation whereif you put out an extra bet and the dealer has a natural,you lose your original bet and the extra bet too.

The dealer winning all on naturals does not affectyour hit/stand decisions. If it is correct to hit your handwhen the dealer takes a hole card, it is also correct to hityour hand if the dealer does not take a hole card. The fact


Table 29Lose All to a Natural, 817


double not a/lowed after splitA-A spl spl spl spl spl -9 -8 -7 -6 h10-10 8 6 5 49-9 0 -2 -3 -4 -4 6 -8 -98-8 spl spl spl spl spl spl spl spl h/- h/-7-7 -9 spl spl spl spl spl h h h h6-6 2 0 -3 -5 -7 h h h h h5-5 never split - play as total of ten4-4 never split - play as total of eight3-3 8 3 0 -2 -9 spl h h h h2-2 7 3 0 -4 -9 spl h h h h



11 db db db db db -9 -6 -4 3 h10 -8 -9 -10 db db -6 -4 -1 h h9 1 0 -2 -4 -6 3 7 h h h8** h 9 5 3 1 h h h h h7 h h h 9 9 h h h h h6 h h h h h h h h h h5 h h h h h h h h h h

No HOLE CARD 83


double allowed after splitA-A spl spl spl spl spl -9 -8 -7 -6 h10-10 8 6 5 49-9 -2 -3 -5 -6 -6 3 -8 -98-8 spl spl spl spl spl spl spl spl h/- h/-7-7 -9 spl spl spl spl spl 5 h h h6-6 -1 -4 -6 -8 -10 h h h h h5-5 never split - play as total of ten4-4** h 6 1 -1 -4 h h h h h3-3 0 -4 -7 -9 spl spl 4 h h h2-2 -3 -5 -7 -9 spl spl 5 h h h

KEY

db

h

h/-

spl

s17

number

**

Stand.

Double down.

Hit.

See elsewhere in the table for hit or stand.

Split.





that the next card out of the shoe could go to either youor the dealer does not affect the play of your hand.

The dealer winning all on naturals also does notaffect the play of your hand if the dealer shows otherthan a 10 or ace. The tables in this chapter have colwnnsfor dealer showing 2-9, but those columns are identicalto those in table 9 (page 46).

Basic strategy, if you lose all to a natural by thedealer, is to split ace-ace against 10 but otherwise do notdouble down or split against 10 or ace.

Advice for card counters if the dealer might have anatural to beat your doubled bet is contained in table 29.Note that the only doubling down to consider against a10 or ace is with eleven against 10. The only splitting toconsider against a 10 or ace is ace-ace against 10.

The strategy numbers for playing against 10 andace do not depend on whether you can double down aftersplitting.

The advice to never double or split against an aceholds whether the dealer hits or stands on soft seventeen. Table 29 assumes the dealer stands on soft seventeen. If you are playing against the lose-all rule in acasino where the dealer hits soft seventeen, then use the10 and ace columns of table 29 with columns for otherupcards from table 10 (page 48).

Having to pass up those double downs and splitsagainst lOs and aces is costly; simulation of losing all to

Lose all to a naturalWin rate: $11Std dev: $411

No HOLE CARD 85

a natural by the dealer yields a win rate of $5 less thanthe benchmark.

Table 30 summarizes the amendments to thoseparts of the appendix that deal with rules for doublingdown and pair splitting against 10 or ace when you loseall to a dealer's natural. (The appendix contains strategyindexes for the high-low and halves counts for one andmultiple decks and both dealer actions on soft seventeen.) No other decision is affected by whether or not thedealer takes a hole card.

Any 21 Ties Dealer NaturalThis section applies if the dealer's two-card twenty

one counts as an ordinary twenty-one and not as anatural winner. You should be slightly more likely to hit.

Table 30Modifications to Appendixes

When You Lose All to a Natural

Table

AlA2A3A4BlB2B3B4

DoubleElevenvs 10

54544343

SplitAce-Ace

vs 10

-9-6-9-6

-10-6

-10-6

SplitAce-Acevs Ace

1624172416231723


Table 31No Dealer Natural, S17


double not allowed after splitA-A spl spl spl spl spl -9 -8 -7 -8 h10-10 8 6 5 49-9 0 -2 -3 -4 -4 6 -8 -98-8 spl spl spl spl spl spl spl spl h/- h7-7 -9 spl spl spl spl spl h h h h6-6 2 0 -3 -5 -7 h h h h h5-5 never split - play as total of ten4-4 never split - play as total of eight3-3 8 3 0 -2 -9 spl h h h h2-2 7 3 0 -4 -9 spl h h h h

ace-9 10 8 6 5 4ace-8 8 5 3 1 1ace-7 -/db 0 -2 -6 -8 -10 h h h/-ace-7 bI- h h 1ace-6 1 -3 -7 -10 db h h h h hace-5 h 4 -2 -6 db h h h h hace-4 h 7 0 -4 -9 h h h h hace-3 h 7 1 -1 -4 h h h h hace-2 h 7 3 0 -1 h h h h h

hard 20hard 19hard 18hard 17hard 16 -8 -10 9 7 5 1 hhard 15 -5 -6 -7 -9 -9 10 10 8 5 hhard 14 -3 -4 -6 -7 -7 h h h h hhard 13 0 -1 -3 -4 -4 h h h h hhard 12 3 2 0 -1 0 h h h h h

11 db db db db db -9 -6 -4 0 h10 -8 -9 -10 db db -6 -4 -1 h h9 1 0 -2 -4 -6 3 8 h h h8** h 9 5 3 1 h h h h h7 h h h 9 9 h h h h h6 h h h h h h h h h h5 h h h h h h h h h h

No HOLE CARD 87


double allowed after splitA-A spl spl spl spl spl -9 -8 -7 -8 h10-10 8 6 5 49-9 -2 -3 -5 -6 -6 3 -8 -98-8 spl spl spl spl spl spl spl spl h/- h7-7 -9 spl spl spl spl spl 5 h h h6-6 -1 -4 -6 -8 -10 h h h h h5-5 never split - play as total of ten4-4** h 6 1 -1 -4 h h h h h3-3 0 -4 -7 -9 spl spl 4 h h h2-2 -3 -5 -7 -9 spl spl 5 h h h

KEY

db

h

h/-

spl

s17

number

**

Stand.

Double down.

Hit.


Split.





The appropriate advice is shown in table 31. Table 31 isthe same as table 9 (page 46) except for the 10 and acecolumns.

Table 31 assumes you can tie all the dealer's twocard twenty-ones by reaching twenty-one yourself inany number of cards. If your three-or-more-card twentyone ties the dealer's two-card twenty-one for only someupcards and not for others, then use part of table 31 andpart of table 9. For example, you might find a casinowhere the dealers do not check hole cards under 10s butdo check hole cards under aces. Ace-up naturals areturned up immediately, so you never get a chance to geta three-card twenty-one if the dealer shows an ace. Inthat case, use the 10 column from table 31 and the acecolumn from table 9. The other columns are the same onboth tables.

The ace column of table 31 applies whether thedealer stands or hits with soft seventeen. Thus for acasino in which the dealer hits soft seventeen, the 10 andace columns of table 31 can be combined with the othercolumns from table 10 (page 48).

When the dealer with a 10 up and ace for the secondcard has an ordinary twenty-one and not a natural, thewin rate is better than the benchmark by $8 per hour. Sothis rule is quite valuable for the card counter. At thetime of this writing, this rule is offered at several casinosin the Reno area.

Any 21 tiesdlr 10-up natural


89

CHAPTER 8SURRENDER

Surrender is giving up half your bet and not havingto play out the hand. Surrender is a good play only ifthere is no other way to play your hand that has anexpected win of better than -0.5. In other words, sincesurrender means losing half a bet, any other play that onaverage loses less than half a bet is superior to surrender.

The tables in this chapter have two columns fordealer ace. The column labeled "A-s17" is for use whenthe dealer stands on soft seventeen, and the columnlabeled "A-hI7" applies when the dealer hits soft seventeen. Whether the dealer hits or stands on soft seventeendoes not matter when the dealer starts with 7 through10.

Surrender can be worthwhile if the dealer shows 8,9, 10, or ace. When the dealer starts with 2 through 7, youshould not surrender. For a complete set of strategy


numbers when surrender is offered, combine the appropriate table of this chapter with the 2-7 columns of theappropriate table of another chapter, such as chapter 3.

The tables of this chapter break 8-8 from otherplayer hands of sixteen because splitting 8-8 is anattractive alternative. There is no reason to separate 7-7from other fourteens, because for upcards you mightsurrender against you should not split 7-7.

Early SurrenderAt some casinos, you have the option of early surren

der; that is, you may surrender before the dealer checksthe hole card. The higher the count per deck, the morelikely you should be to surrender. Table 32 summarizesearly-surrender decisions. Do not surrender against 27.

Examples:A) You get 9-7 for sixteen against ace; table 32

says always surrender.B) You get 8-8 against 8; table 32 says do not

surrender.C) You get 10-4 for fourteen against 9; surrender

if the count per deck is +6 or more, and do notsurrender if the count per deck is less than +6.

If you lose all to a natural by the dealer and thus willnot split 8-8 against 10, then for 8-8 against lOuse -5, thesame number as for other sixteens against 10.

Early surrender is quite profitable for the cardcounter, adding $19 or $23 to the benchmark win rate,with the amount depending on whether the dealer hitsor stands on soft seventeen. Unfortunately, early surrender is also rare; late surrender is the common form ofsurrender.

SURRENDER

Table 32Early Surrender

91

KEY

blan~

esr

h17

s17

number

Player's Hand Dealer's Upcard8 9 10 A-s17 A-h17

hard 17 5 esr esr10-6, 9-7 4 0 -5 esr esr

8-8 7 -2 esr esrhard 15 7 2 -2 esr esrhard 14 6 0 -9 esrhard 13 3 -7 esrhard 12 8 -4 -8

8 97 -2 -66 -1 -45 0 -24 2 0

Do not surrender.

Surrender.



Surrender if the count per deck equals or exceeds thenumber; do not surrender if the count per deck is lessthan the number.

Early surrender, s17Win rate: $39Std dey: $402

Early surrender, hI7Win rate: $35Std dey: $402


Late SurrenderAt some casinos you have the option of late surren

der. The higher the count per deck, the more likely youshould be to surrender. Table 33 summarizes latesurrender decisions.

Examples:A) You get hard thirteen against ace; table 33

says do not surrender.B) You get 10-4 for fourteen against 9; surrender

if the count per deck is +6 or more, and do notsurrender if the count per deck is less than +6.If you lose all to a natural by the dealer and thus will

not split 8-8 against 10 or ace, then for 8-8 against 10 orace use the same numbers as for other sixteens against10 or ace.

Late surrender is not as profitable as early surrender, of course, since you cannot surrender against anatural. But it is profitable, adding $5 to the benchmarkwin rate.

SURRENDER

Table 33Late Surrender

93


hard 1710-6, 9-7 4 0 -2 -1 -4

8-8 7 0hard 15 7 2 0 2 0hard 14 6 3 6 4hard 13 8

KEY

blank Do not surrender.

h 17 Dealer hits soft seventeen.


number Surrender if the count per deck equals or exceeds thenumber; do not surrender if the count per deck is lessthan the number.

Late surrender, s17Win rate: $21Std dev: $407

Late surrender, h17Win rate: $17Std dev: $408


Table 34Multiple-Card Early Surrender

Player's Hand Dealer's Upcard8 9 iO A-si7 A-hi7

hard 17 5 esr esr10-6, 9-7 4 0 -5 esr esr

8-8 7 -2 esr esrhard 15 7 2 -2 -9 esrhard 14 7 0 -7 esrhard 13 5 -4 -7hard 12 0 -2

8 77 0 -46 0 -15 2 04 5 3

KEY

blank

esr

h17

s17

number

Do not surrender.

Surrender.



Surrender if the count per deck equals or exceeds thenumber; do not surrender if the count per deck is lessthan the number.

Multiple-Card SurrenderAt a few casinos, primarily in Asia, you can surren

der any number of cards. For example, if you start with

Multiple-cardearly surrender


SURRENDER 95

Table 35Multiple-Card Late Surrender


hard 1710-6, 9-7 4 0 -2 -1 -4

8-8 7 0hard 15 7 2 0 2 0hard 14 7 3 6 5hard 13 9

KEY

blank Do not surrender.


s I 7 Dealer stands on soft seventeen.

number Surrender if the count per deck equals or exceeds thenumber; do not surrender if the count per deck is lessthan the number.

6-2 and hit an 8 for sixteen, you can surrender yoursixteen. You are less likely to surrender a hand with alow total if you will still have the option of surrenderingafter one more card.

The strategy numbers for multiple-card early surrender are shown in table 34.

The strategy numbers for multiple-card late surrender are shown in table 35.

Multiple-cardlate surrender



Surrender After Doubling DownAt some casinos in Asia you were allowed to surren

der after doubling down. No strategy table is providedfor this rule because except for seventeen against ace,the best play is simple: After doubling down, surrenderall hands of sixteen or less if the dealer shows 8, 9, 10, orace.

If you have doubled down and ended up withseventeen and the dealer shows an ace, you might wantto surrender. Follow the advice for seventeen against acein the appropriate table from earlier in this chapter.

When to double, if you can surrender after doublingdown, is explained in the chapter on doubling down.

Surrender afterdoubling down


Second-Chance 21A rarely-offered form of surrender is to give up your

hand plus half your bet and receive a new hand. Thedifference between this second-chance option and ordinary surrender is you must play a new hand for theremaining half of your bet rather than simply putting

SURRENDER 97

those chips back on your stack. If the dealer shows a cardthat is powerful enough to make surrender worth considering, then the dealer is showing a card you wouldrather not play against. Therefore, basic strategy saysignore the second-chance option.

If you are counting cards and the count gets highenough, you sometimes are better off giving up half yourbet to get a new hand. The situations arise when youhave hard thirteen to seventeen and the dealer shows 7to 9. How high the count per deck must be to justifygiving up half your bet to get a new hand is shown intable 36. This table applies to both dealer actions on softseventeen.

Table 36Second-Chance 21


hard 17 8hard 16 3 4 5hard 15 4 4 5hard 14 5 8 9hard 13 9

KEY

blank Ignore the second-chance option.

number Give up half the bet to replace the cards if the countper deck equals or exceeds the number; play out theoriginal hand if the count per deck is less than thenumber.


CHAPTER 9

FIVE CARDS

Though this chapter is titled "Five Cards," it alsocovers bonuses for hands of six and seven cards. Thestrategy for playing them depends on the size of thebonus and the number of additional cards required to beeligible for the bonus.

The tables in this chapter tell you when you shouldhit, given the number of additional number of cards youneed to qualify for the bonus. The more cards already inyour hand the closer you are to the bonus, and thus themore likely you are to hit. Of course you stop hitting andsmile when you have enough cards to qualify for thebonus.

Every live-game bonus I have seen has been optional, and if you accept it the hand is over and the dealerpicks up your cards. (Some blackjack slot machines havea bonus that is paid automatically because no morecards will fit on the screen.)

FIVE CARDS

Half a Bet

99

If the bonus is small, you might be better off refusingit. For example, Macao has a bonus of half a bet for a fivecard hand. With a bonus that small, some five-cardhands are worth more when played out to maybe win,maybe push, maybe lose. Tables 37-44 are for use whenthe bonus is half a bet.

If you happen to be offered a half-win bonus for sixcards, then use tables 38-40 and 42-44 for hands of onemore card; for example use table 40 for six-card handsinstead of five-card hands.

Tables 37-44 assume the dealer stands on softseventeen. They also assume you can double down onany first two cards, but cannot double down on morethan two cards. They also assume you lose only one betif the dealer has a natural.

The casinos that I know have offered this rule all arelocated in Asia. At all of them, dealers stand on softseventeen. If you need five-card half-win tables fordealers hit soft seventeen, you can generate them yourself with Blackjack Count Analyzer.

Tables 37-40 assume you cannot collect a half win ifthe dealer has a natural.



Table 372-Card Hands

With 5-Card Half Win, 817Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 A

double not allowed after splitA-A -7 -8 -9 -9 -10 -6 -3 -1 -1 010-10 8 6 5 49-9 0 -2 -3 -4 -4 6 -8 -9 38-8 spl spl. spl spl spl spl spl spl 6* spl7-7 -9 spl spl spl spl spl 5 h h h6-6 2 0 -3 -5 -7 h h h h h5-5 never split - play as total of ten4-4 never split - play as total of eight3-3 9 5 2 0 -2 h h h h h2-2 10 6 3 0 0 h h h h h

ace-9 10 8 6 5 4ace-8 8 5 3 1 1ace-7 -/db 0 -2 -6 -8 -10 h h h h/-ace-7 bI- h h h 3ace-6 2 -2 -5 -8 db h h h h hace-5 h 5 -1 -4 -9 h h h h hace-4 h 9 2 -1 -5 h h h h hace-3 h 9 4 1 -1 h h h h hace-2 h 10 6 3 1 h h h h hhard 20hard 19hard 18hard 17 -6hard 16 -8 -10 9 7 5 0 8hard 15 -5 -6 -7 -9 -9 10 10 8 4 10hard 14 -3 -4 -6 -7 -7 h h h h hhard 13 0 -1 -3 -4 -4 h h h h hhard 12 3 2 0 -1 0 h h h h h11 db db db db db -8 -6 -3 -3 210 -8 -9 db db db -5 -3 0 5 49 1 0 -2 -4 -6 4 8 h h h8** h 9 5 3 1 h h h h h7 h h h 9 9 h h h h h6 h h h h h h h h h h5 h h h h h h h h h h

FIVE CARDS 101


double allowed after splitA-A -7 -8 -9 -9 -10 -6 -3 -1 -1 010-10 8 6 5 49-9 -2 -3 -5 -6 -6 3 -8 -9 38-8 spl spl spl spl spl spl spl spl 8* spl7-7 -9 spl spl spl spl spl 5 h h h6-6 -1 -4 -6 -8 -10 h h h h h5-5 never split - play as total of ten4-4** h 6 1 -1 -4 h h h h h3-3 2 0 -2 -4 -7 -4 h h h h2-2 4 1 -1 -3 -5 1 h h h h

KEY

Five-card half winWin rate: $30Std dev: $412

db

h

h/-

spl

s17

number

*

**

Stand.

Double down; if you cannot double, then hit.

Hit.


Split.


Stand (or double or split) at a count per deck equal toor greater than the number; hit (or do not split) at acount per deck below the number. (Note that the 3for ace-7 against ace is hit/stand, and does not say todouble down.)

Reverse the meaning. Hit (or do not split) at a countper deck equal to or greater than the number; stand(or double or split) at a count per deck less than thenumber.

With 4-4, sptitting isbetter than doublingdown if doubling isallowed after splitting.




soft 21soft 20soft 19 10*soft 18 2 0 0 -6 -6 6 h h hsoft 17 h h h h h h h h h hsoft 16 h h h h h h h h h hsoft 15 h h h h h h h h h hsoft 14 h h h h h h h h h hsoft 13 h h h h h h h h h h

hard 20hard 19hard 18hard 17 -6hard 16 -8 -10 9 7 5 0 8hard 15 -5 -6 -7 -9 -9 h h 10 6 hhard 14 -3 -4 -6 -7 -7 h h h h hhard 13 0 -1 -3 -4 -4 h h h h hhard 12 4 3 1 0 0 h h h h h

KEY

h

s17

number

*

Stand.

Hit.


Stand at a count per deck equal to or greater thanthe number; hit at a count per deck less than thenumber.

Reverse the meaning. Hit at a count per deck equalto or greater than the number; stand at a count perdeck less than the number.

*

FIVE CARDS 103


With 5-Card Half Win, S17Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 A

soft 21soft 20 1*soft 19 h h h h h -5 -2 h h hsoft 18 h h h h h h h h h hsoft 17 h h h h h h h h h hsoft 16 h h h h h h h h h hsoft 15 h h h h h h h h h hsoft 14 h h h h h h h h h hsoft 13 h h h h h h h h h h

hard 20hard 19hard 18hard 17 -2hard 16 -4 -5 -7 -8 -9 h h h h hhard 15 1 0 -1 -3 -3 h h h h hhard 14 6 4 2 1 0 h h h h hhard 13 h 9 7 5 5 h h h h hhard 12 h h h 9 10 h h h h h

KEY

Stand.

h IIit.


number Stand at a count per deck equal to or greater thanthe number; hit at a count per deck less than thenumber.

Reverse the meaning. IIit at a count per deck equalto or greater than the number; stand at a count perdeck less than the number.




soft 21 rej rej rej rej rej rej rej rej rej rejsoft 20 rej rej rej rej rej rej rej rej 10· rejsoft 19 bon bon bon 8 1 rej rej bon bon bonsoft 18 bon bon bon bon bon 10 bon bon bon bonsoft 17 bon bon bon bon bon bon bon bon bon bonsoft 16 bon bon bon bon bon bon bon bon bon bonsoft 15 bon bon bon bon bon bon bon bon bon bon

hard 21 rej rej rej rej rej rej rej rej rej rejhard 20 rej rej rej rej rej rej rej rej 10* rejhard 19 bon bon bon 8 1 rej rej bon bon bonhard 18 bon bon bon bon bon 10 bon bon bon bonhard 17 bon bon bon bon bon bon bon bon bon bonhard 16 bon bon bon bon bon bon bon bon bon bonhard 15 bon bon bon bon bon bon bon bon bon bonhard 14 bon bon bon bon bon bon bon bon bon bonhard 13 bon bon bon bon bon bon bon bon bon bonhard 12 bon bon bon bon bon bon bon bon bon bon

II bon bon bon bon bon bon bon bon bon bon10 bon bon bon bon bon bon bon bon bon bon

KEY

bonrej

s17

number

*

Accept bonus of half a bet.

Reject bonus of half a bet, and stand.


Reject bonus and stand at a count per deck equal toor greater than the number; accept bonus at a countper deck less than the number.

Reverse the meaning. Accept bonus at a count perdeck equal to or greater than the number; rejectbonus and stand at a count per deck less than thenumber.

FIVE CARDS 105

At some casinos that offer the five-card half win,dealers do not take hole cards and you lose allan doublesor splits if the dealer has a natural. At those casinos, ifyou get a five-card hand you can collect your half-winbefore you fmd out if the dealer has a natural. This bookcalls this "early half-win." Tables 41-44 assume youcollect your half-win before the dealer takes a secondcard, or if one is taken, then before the dealer checks it.The columns for 2-9 are identical to tables 37-40 sincethe dealer cannot get a natural except with a 10 or aceup.

If you are playing the five-card half win at a casinowhere the dealer checks under the ace but not under the10, then use the ace columns from tables 37-40 and the10 columns from tables 41-44.



Table 41Play of 2-Card Hands With5-Card Early Half Win, S17


double not allowed after splitA-A -7 -8 -9 -9 -10 -6 -3 -1 2 h10-10 8 6 5 49-9 0 -2 -3 -4 -4 6 -8 -98-8 spl spl spl spI spl spl spl spl h/- h/-7-7 -9 spI spl spI spl spI h h h h6-6 2 0 -3 -5 -7 h h h h h5-5 never split - play as total of ten4-4 never split - play as total of eight3-3 9 5 2 0 -2 h h h h h2-2 10 6 3 0 0 h h h h h

ace-9 10 8 6 5 4ace-8 8 5 3 1 1ace-7 -/db 0 -2 -6 -8 -10 h h h h/-ace-7 bI- h h h 8ace-6 2 -2 -5 -8 db h h h h hace-5 h 5 -1 -4 -9 h h h h hace-4 h 9 2 -1 -5 h h h h hace-3 h 9 4 1 -1 h h h h hace-2 h 10 6 3 1 h h h h h

hard 20hard 19hard ] 8hard 17 -6hard 16 -8 -10 9 7 5 0 8hard 15 -5 -6 -7 -9 -9 10 10 8 4 10hard 14 -3 -4 -6 -7 -7 h h h h hhard ]3 0 -1 -3 -4 -4 h h h h hhard 12 3 2 0 -1 0 h h h h h

1] db db db db db -8 -6 -3 6 h10 -8 -9 db db db -5 -3 0 h h9 1 0 -2 -4 -6 4 8 h h h8** h 9 5 3 1 h h h h h7 h h h 9 9 h h h h h6 h h h h h h h h h h5 h h h h h h h h h h

FIVE CARDS 107


double allowed after splitA-A -7 -8 -9 -9 -10 -6 -3 -1 2 h10-10 8 6 5 49-9 -2 -3 -5 -6 -6 3 -8 -98-8 spl spl spl spl spl spl spl spl h/- h/-7-7 -9 spl spl spl spl spl 5 h h h6-6 -1 -4 -6 -8 -10 h h h h h5-5 never split - play as total of ten4-4** h 6 1 -1 -4 h h h h h3-3 2 0 -2 -4 -7 -4 h h h h2-2 4 1 -1 -3 -5 1 h h h h

KEY

db

h

h/-spl

s17

number

**

Stand.


Hit.


Split.




Five-card early half winWin rate: $36Std dey: $411


Table 423-Card Hands With 5-Card

Early Half Win, 817Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 A

soft 21soft 20soft 19 1* hsoft 18 2 0 0 -6 -6 6 h h hsoft 17 h h h h h h h h h hsoft 16 h h h h h h h h h hsoft 15 h h h h h h h h h hsoft 14 h h h h h h h h h hsoft 13 h h h h h h h h h h

hard 20hard 19hard 18hard 17 -6hard 16 -8 -10 9 7 5 0 8hard 15 -5 -6 -7 -9 -9 h h 10 6 hhard 14 -3 -4 -6 -7 -7 h h h h hhard 13 0 -1 -3 -4 -4 h h h h hhard 12 4 3 1 0 0 h h h h h

KEY

h

s17

number

*

Stand.

Hit.


Stand at a count per deck equal to or greater thanthe number; hit at a count per deck less than thenumber.

Reverse the meaning. Hit at a count per deck equalto or greater than the number; stand at a count perdeck less than the number.

FIVE CARDS 109



soft 21 hsoft 20 h hsoft 19 h h h h h -5 -2 h h hsoft 18 h h h h h h h h h hsoft 17 h h h h h h h h h hsoft 16 h h h h h h h h h hsoft 15 h h h h h h h h h hsoft 14 h h h h h h h h h hsoft 13 h h h h h h h h h h

hard 20hard 19hard 18hard 17 hhard 16 -4 -5 -7 -8 -9 h h h h hhard 15 1 0 -1 -3 -3 h h h h hhard 14 6 4 2 1 0 h h h h hhard 13 h 9 7 5 5 h h h h hhard 12 h h h 9 10 h h h h h

KEY

Stand.

h IIit.






soft 21 rej rej rej rej rej rej rej rej rej bonsoft 20 rej rej rej rej rej rej rej rej -8* bonsoft 19 bon bon bon 8 1 rej rej bon bon bonsoft 18 bon bon bon bon bon 10 bon bon bon bonsoft 17 bon bon bon bon bon bon bon bon bon bonsoft 16 bon bon bon bon bon bon bon bon bon bonsoft 15 bon bon bon bon bon bon bon bon bon bon

hard 21 rej rej rej rej rej rej rej rej rej bonhard 20 rej rej rej rej rej rej rej rej bon bonhard 19 bon bon bon 8 1 rej rej bon bon bonhard 18 bon bon bon bon bon 10 bon bon bon bonhard 17 bon bon bon bon bon bon bon bon bon bonhard 16 bon bon bon bon bon bon bon bon bon bonhard 15 bon bon bon bon bon bon bon bon bon bonhard 14 bon bon bon bon bon bon bon bon bon bonhard 13 bon bon bon bon bon bon bon bon bon bonhard 12 bon bon bon bon bon bon bon bon bon bon

11 bon bon bon bon bon bon bon bon bon bon10 bon bon bon bon bon bon bon bon bon bon

KEY

bon

rej

s17

number

*

Accept bonus of half a bet.

Reject bonus of half a bet, and stand.


Reject bonus and stand at a count per deck equal toor greater than the number; accept bonus at a countper deck less than the number.Reverse the meaning. Accept bonus at a count perdeck equal to or greater than the number; rejectbonus and stand at a count per deck less than thenumber.

FIVE CARDS 111


Automatic Win, 817Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 A

soft 21soft 20soft 19soft 18 -10 -8 h h 1soft 17 h h h h h h h h h hsoft 16 h h h h h h h h h hsoft 15 h h h h h h h h h hsoft 14 h h h h h h h h h hsoft 13 h h h h h h h h h h


KEY

Stand.

h IIit.



Six-card automatic winWin rate: $18Std dev: $415



Automatic Win, S17Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 Asoft 21soft 20soft 19 h -6soft 18 h h h h 8 -1 h h h 5soft 17 h h h h h h h h h hsoft 16 h h h h h h h h h hsoft 15 h h h h h h h h h hsoft 14 h h h h h h h h h hsoft 13 h h h h h h h h h h

hard 20hard 19hard 18hard 17 -6hard 16 -8 -10 9 7 5 1 8hard 15 -4 -6 -7 -8 -9 h h h 8 10hard 14 -1 -3 -4 -5 -6 h h h h hhard 13 2 0 0 -2 -2 h h h h hhard 12 7 5 3 1 2 h h h h h

KEY

Stand.

h IIit.



FIVE CARDS 113


Automatic Win, S17Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 A

soft 21 h h h h h h h h h hsoft 20 h h h h h h h h h hsoft 19 h h h h h h h h h hsoft 18 h h h h h h h h h hsoft 17 h h h h h h h h h hsoft 16 h h h h h h h h h hsoft 15 h h h h h h h h h hsoft 14 h h h h h h h h h hsoft 13 h h h h h h h h h h

hard 20hard 19hard 18hard 17 0 h h 7hard 16 3 1 0 -1 -2 h h h h hhard 15 8 6 4 3 3 h h h h hhard 14 h h 9 7 8 h h h h hhard 13 h h h h h h h h h hhard 12 h h h h h h h h h h

KEYStand.

h IIit.




Table 482-Card Hands With db3, 6-Card

Win, HI7Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 A

double not allowed after splitA-A -9 -10 -10 spl spl -7 -6 -5 -5 -210-10 8 6 5 49-9 -1 -2 -3 -4 -6 6 -8 -9 28-8 spl spl spl spl spl spl spl spl 6* -17-7 -8 -10 spl spl spl spl h h h h6-6 1 -1 -3 -5 -7 h h h h h5-5 never split - play as total of ten4-4** never split - play as total of eight3-3 h 9 5 2 0 0* h h h h2-2 h 9 5 2 0 h h h h h

ace-9 10 8 6 5 4ace-8 8 5 3 1 0ace-7 -/db 0 -2 -6 -8 db h h hace-7 hI- h h hace-6 2 -3 -7 -10 db h h h h hace-5 h h 0 -5 db h h h h hace-4 h h 5 -1 -7 h h h h hace-3 h h 8 2 -1 h h h h hace-2 h h 9 4 2 h h h h h

hard 20hard 19hard 18hard 17 -4hard 16 -9 -10 9 7 5 0 3hard 15 -5 -7 -8 -9 10 10 8 4 5hard 14 -3 -5 -6 -7 -9 h h h h 9hard 13 0 -2 -3 -4 -7 h h h h hhard 12 3 1 0 -1 -3 h h h h h11 db db db db db -9 -6 -4 -4 010 -8 -9 -10 db db -6 -4 -1 4 39 2 0 -1 -2 -4 4 9 h h h8** h h h 9 8 h h h h h7 h h h h h h h h h h6 h h h h h h h h h h5 h h h h h h h h h h

FIVE CARDS 115

Table 48 continuedPlayer's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 A

double allowed after splitA-A -9 -10 -10 spl spl -7 -6 -5 -5 -210-10 8 6 5 49-9 -2 -3 -5 -6 -8 3 -8 -9 18-8 spl spl spl spl spl spl spl spl 8* -17-7 -9 spl spl spl spl spl 5 h h h6-6 -2 -4 -6 -8 -10 h h h h h5-5 never split - play as total of ten4-4** h h 8 3 0 h h h h h3-3 3 0 -2 -4 -6 spl h h h h2-2 4 0 -2 -4 -6 -5 h h h h

KEY

-/db

db

h

hI7

h/-

spl

number

*

**

Stand.



Hit.



Split.





Table 493-Card Hands With db3, 6-Card

Win, HI7Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 A

soft 21soft 20 10 8 6 5 4soft 19 8 5 3 1 0soft 18 -/db 0 -2 -6 -8 db h/- h h hsoft 18 hI- -9 -10 -10 s/db s/db -8 h h hsoft 17 5 0 -3 -6 -10 h h h h hsoft 16 h h 2 -1 -6 h h h h hsoft 15 h h 8 2 -1 h h h h hsoft 14 h h 9 5 3 h h h h hsoft 13 h h h 7 6 h h h h h

hard 20hard 19hard 18hard 17 -4hard 16 -9 -10 9 7 5 0 3hard 15 -5 -7 -8 -9 10 10 8 4 5hard 14 -3 -5 -6 -7 -9 h h h h 9hard 13 0 -2 -3 -4 -7 h h h h hhard 12 3 1 0 -1 -3 h h h h h

11 db db db db db -8 -5 -2 -2 110 -7 -8 -9 db db -5 -2 0 7 49 3 0 0 -2 -4 5 9 h h h8 h h h 9 8 h h h h h7 h h h h h h h h h h

KEYStand.

-/db Do not hit. See the -/db row for the number saying whether tostand or double down.

db Double down; if you cannot double, then hit.

h Hit.

hI7 Dealer hits soft seventeen.

h/- Do not double. See the h1- row for the number saying whether tohit or to stand.

number Stand or double at a count per deck equal to or greater than thenumber; hit (or do not double) at a count per deck less than thenumber.

FIVE CARDS 117

Table 504-Card Hands With 6-Card Win,

HI7Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 A

soft 21soft 20 10 8 6 5 4soft 19 8 5 3 1 0 h 0soft 18 h h h h h -1 h h h hsoft 17 h h 10 7 5 h h h h hsoft 16 h h h h h h h h h hsoft 15 h h h h h h h h h hsoft 14 h h h h h h h h h h

hard 20hard 19hard 18hard 17 -4hard 16 -9 -10 9 7 5 1 3hard 15 -5 -6 -7 -8 h h h 8 7hard 14 -2 -3 -4 -6 -8 h h h h hhard 13 1 0 -1 -2 -4 h h h h hhard 12 6 4 3 1 0 h h h h h

11 h h h h h h h h h h10 h h h h h h h h h h9 h h h h h h h h h h8 h h h h h h h h h h

KEY

Stand.

h IIit.


number Stand or double at a count per deck equal to orgreater than the number; hit (or do not double) at acount per deck less than the number.



With 6-Card Win, Ht7Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 Asoft 21 h h h h h h h h h hsoft 20 h h h h h h h h h hsoft 19 h h h h h h h h h hsoft 18 h h h h h h h h h hsoft 17 h h h h h h h h h hsoft 16 h h h h h h h h h hsoft 15 h h h h h h h h h h

hard 20hard 19hard 18hard 17 0 h h 9hard 16 3 1 0 -2 -3 h h h h hhard 15 8 6 4 3 2 h h h h hhard 14 h h 9 7 7 h h h h hhard 13 h h h h h h h h h hhard 12 h h h h h h h h h h11 h h h h h h h h h h10 h h h h h h h h h h

KEY

Stand.

h IIit.



FIVE CARDS

Automatic Win

119

Tables 45-51 cover the situation of six cards is anautomatic winner of the same amount as you have bet.In this case it is always worthwhile accepting the bonus- you should never turn it down.

Note that it is worthwhile hitting soft twenty-one ifone more card will give you an automatic winner. A surewinner is worth more than a hand that will probably winbut might push.

Tables 45-51 assume the automatic winner appliesto six-card hands. If you happen to be offered an automatic win for five cards, then use the tables 45-51 forhands of one less card; for example use table 47 for fourcard hands instead of five-card hands. Likewise if youhappen to be offered an automatic winner for seven-cardhands, then use tables 45-51 for hands of one more card.

Tables 45-51 also assume you lose only one bet if thedealer has a natural. Tables 45-47 assume the dealerstands on soft seventeen, and they assume you candouble down on any first two cards.

Tables 48-51 assume the dealer hits soft seventeenand also assume you can double down on two or threecards. Las Vegas Club in Las Vegas (at this writing) hasthese rules, and allows double down on two cards aftersplitting.

The numbers for soft eighteen in table 48 are forstand versus double. If the count per deck is below thenumber in the table, then stand; if the count per deck isequal to or greater than the number in the table, thendouble.

Soft eighteen is more complicated if: you have it inthree cards, you can double down on three cards, and asix-card hand is an automatic winner. Therefore, table



Table 522-Card Hands With Winning

5-Card 21 Pays 2-1, S17Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 A

double not allowed after splitA-A -10 -10 spl spI spI -8 -6 -6 -6 -210-10 8 6 5 49-9 0 -2 -3 -4 -4 6 -8 -9 38-8 spl spl spl spl spI spI spI spl 6* spl7-7 -9 spl spl spI spl spl h h h h6-6 2 0 -3 -5 -7 h h h h h5-5 never split - play as total of ten4-4 never split - play as total of eight3-3 8 4 0 -1 -4 8 h h h h2-2 7 4 0 -2 -6 0 h h h h



11 db db db db db -9 -6 -4 -4 110 -8 -9 -10 db db -5 -3 0 5 49 1 0 -2 -4 -6 4 8 h h h8** h 9 5 3 1 h h h h h7 h h h 9 9 h h h h h6 h h h h h h h h h h5 h h h h h h h h h h

FIVE CARDS 121


double allowed after splitA-A -10 -10 spl spl spl -8 -6 -6 -6 -210-10 8 6 5 49-9 -2 -3 -5 -6 -7 3 -8 -9 38-8 spl spl spl spl spl spl spl spl 8* spl7-7 -9 spl spl spl spl spl 5 h h h6-6 -1 -4 -6 -8 -10 h h h h h5-5 never split - play as total of ten4-4** h 8 3 0 0 h h h h h3-3 0 -1 -5 -7 spl spl h h h h2-2 0 -3 -5 -7 -10 -6 9 h h h

KEY

-/db

db

h

h/-

spl

s17

number

*

**

Stand.



Hit.


Split.







5-Card 21 Pays 2-1, S17Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 A

soft 21soft 20soft 19soft 18 -10 -10 -10 -8 h h 6soft 17 h h h h h h h h h hsoft 16 h h h h h h h h h hsoft 15 h h h h h h h h h hsoft 14 h h h h h h h h h hsoft 13 h h h h h h h h h h

hard 20hard 19hard 18hard 17 -6hard 16 -8 -10 9 7 5 0 8hard 15 -5 -6 -7 -9 -9 h h 10 5 hhard 14 -3 -4 -6 -7 -7 h h h h hhard 13 0 -1 -3 -4 -4 h h h h hhard 12 3 2 0 -1 0 h h h h h

KEY

Stand.

h I1it.



FIVE CARDS 123


5-Card 21 Pays 2-1, 817Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 A

soft 21soft 20soft 19soft 18 h h h -6 -6 h h h hsoft 17 h h h h h h h h h hsoft 16 h h h h h h h h h hsoft 15 h h h h h h h h h hsoft 14 h h h h h h h h h hsoft 13 h h h h h h h h h h

hard 20hard 19hard 18hard 17 -7 -8 0hard 16 -4 -6 -7 -8 -9 h h h 10 hhard 15 -1 -3 -4 -5 -6 h h h h hhard 14 0 0 -2 -3 -3 h h h h hhard 13 4 2 0 0 0 h h h h hhard 12 8 6 4 3 4 h h h h h

KEY

Stand.

h IIit.




49 has two rows for soft eighteen. One row is for thedouble-down decision, and one row is for the hit/standdecision. Always check the double-down row first; if yourcount per deck is high enough to justify a double down,then do not bother with the hit/stand row.

Double Pay on Five-Card 21Tables 52-54 cover the situation where five cards is

an automatic 2:1 winner only if you get exactly twentyone and the dealer does not also get twenty-one. Thisbonus is less valuable than if you were to get the bonusno matter what the total points in your hand, and nomatter whether that total was a winner.

When the number of cards required for the bonus isfive and you must get exactly twenty-one and the totalmust win, the bonus adds only about $1 per hour to thebenchmark. This bonus has been offered at a few tablesin Atlantic City since 1991.

Double pay on five-card 21Win rate: $17Std dev: $415

125

CHAPTER 106-7-8 ETC

This chapter covers bonuses for 6-7-8 and three 7s.Each table in this chapter is divided into several

sections. The reason is there are several ways that thebonus can interact with surrender. You might be paid thebonus even if the dealer turns out to have a natural onthe hand, or the bonus might be paid only when thedealer does not have a natural. And there are both earlyand late surrender. All of the tables in this chapterassume you are paid the bonus if the dealer achievestwenty-one in more than two cards.


6-7-8The next few sections apply to bonuses for 6-7-8.If the bonus applies only to one suit, such as spades,

the numbers in the sections for 6-7-8 suited apply. Yousimply use these numbers when your first two cards areof the appropriate suit.

Tables 55-58 are for use when the dealer stands onsoft seventeen; tables 59-62 are parallel and apply whenthe dealer hits soft seventeen.

Double Pay for 6-7-8Double for Mixed Suits

In 1990 The Red Lion in Elko paid double for any 67-8; the cards did not have to be of the same suit. Thismade hitting 6-7, 6-8, and 7-8 more attractive. Theappropriate strategy numbers defining where hittingbecomes more attractive than standing are shown intable 55.

Red Lion did not offer surrender. However, table 55also indicates when it is better to hit than to earlysurrender, and when it is better to hit than to latesurrender. Note that hi-means do not surrender; insteaduse the hit-stand part of the table.

Double for any 6-7-8Win rate: $21Std dev: $417

6-7-8 ETC 127

Table 55·Any 6-7-8 Pays Double, S17


hit/standany 7-8 -1 -2 -3 -4 -5 h h h h hany 6-8 1 0 -1 -2 -2 h h h h hany 6-7 5 3 1 0 0 h b b b b

early surrender, assuming 6-7-8 beats a naturalany 7-8 h/- h/- h/- b/- h/- b h 7 3 -1any 6-8 b/- h/- h/- h/- h/- h h h 8 0any 6-7 b/- h/- h/- h/- h/- h h h h 3

early surrender, assuming 6-7-8 loses to a naturalany 7-8 h/- h/- h/- h/- h/- h h 7 2 -6any 6-8 h/- h/- h/- h/- h/- h h h 6 -7any 6-7 b/- h/- h/- h/- h/- h h h h -7

late surrender, assuming 6-7-8 beats a naturalany 7-8 h/- h/- h/- h/- h/- h h 7 6 hany 6-8 h/- h/- h/- h/- h/- h h h b hany 6-7 h/- h/- h/- h/- h/- h h h h h

late surrender, assuming 6-7-8 loses to a naturalany 7-8 h/- h/- h/- h/- h/- b h 7 5 7any 6-8 h/- h/- h/- h/- h/- b h h h hany 6-7 h/- h/- h/- b/- h/- h h h b b

KEY

h flit.

hi- See the first three lines of the table for hit or stand.


number Stand (against 2 through 6) or surrender (against 8,9, 10, or ace) if the count per deck equals or exceedsthe number; bit if tbe count per deck is less than thenumber.


Double For Same SuitMany casinos have offered double pay for 6-7-8 of

the same suit. That is not nearly as valuable as doublepay for any 6-7-8. And, since chances of catching the cardneeded for the bonus are only 1/4 as high as if mixedsuits would earn the bonus, hitting suited 6-7 or 6-8 or7-8 is not as attractive as if any 6-7-8 paid double. Table56 indicates when it is better to hit than to stand, whenit is better to hit than to early-surrender, and when it isbetter to hit than to late-surrender.

Double for suited 6-7-8Win rate: $16Std dey: $415

6-7-8 ETC 129

Table 566-7-8 Suited Pays Double, S17

Player'sHand 2 3

hit/stand7-8 ss -4-56-8 ss -2-36-7 ss 1 0

Dealer's Upcard45678

-6 -7 -8 h h-4 -6 -6 h h-1 -3 -3 h h

9 10 A

10 7 hh h hh h h

early surrender, assuming 6-7-8 suited beats a natural7-8 ss h/- h/- h/- h/- h/- h 8 4 0 -96-8 ss h/- h/- h/- h/- h/- h h 8 1 -96-7 ss hI- hI- hI- hI- hI- h h h 5 -7

early surrender, assuming 6-7-8 suited loses to a natural7-8 ss hI- hI- hI- hi- hI- h 8 4 -1 -1 06-8 ss h/- h/- h/- h/- h/- h h 8 1 -96-7 ss hI- hi- hi- hi- hI- h h h 5 -7

late surrender, assuming 6-7-8 suited beats a natural7-8 ss h/- h/- h/- h/- h/- h 8 4 1 36-8 ss h/- h/- h/- h/- h/- h h 8 5 86-7 ss hi- hi- hi- hI- hi- h h h 10 h

late surrender, assuming 6-7-8 suited loses to a natural7-8 ss h/- h/- h/- h/- h/- h 8 4 1 36-8 ss h/- h/- h/- h/- h/- h h 8 5 86-7 ss hI- hi- hi- hi- hi- h h h 10 h

KEYh lIit.

hi- See the fIrst three lines of the table for hit or stand.

s s Same suit.


number Stand (against 2 through 6) or surrender (against 8,9, 10, or ace) if the count per deck equals or exceedsthe number; hit if the count per deck is less than thenumber.


Triple Pay For 6-7-8Triple for Mixed Suits

I know of no casino that has offered triple pay for 67-8 of mixed suits. However, should you encounter thisrule, table 57 indicates when it is better to hit than tostand, when it is better to hit than to early-surrender,and when it is better to hit than to late-surrender.

Triple for any 6-7-8Win rate: $25Std dev: $418

6-7-8 ETC 131

Table 57Any 6-7-8 Pays Triple, S17


hit/standany 7-8 2 0 0 -1 -1 h h h h hany 6-8 5 4 2 0 1 h h h h hany 6-7 10 7 6 4 5 h h h h h

early surrender, assuming 6-7-8 beats a naturalany 7-8 h/- h/- h/- h/- h/- h h h 7 4any 6-8 h/- h/- h/- h/- h/- h h h h 10any 6-7 h/- h/- h/- h/- h/- h h h h h

early surrender, assuming 6-7-8 loses to a naturalany 7-8 h/- h/- h/- h/- h/- h h h 6 -2any 6-8 h/- h/- h/- h/- h/- h h h h -1any 6-7 h/- h/- h/- h/- h/- h h h h 0

late surrender, assuming 6-7-8 beats a naturalany 7-8 h/- h/- h/- h/- h/- h h h h hany 6-8 h/- h/- h/- h/- h/- h b h h bany 6-7 h/- h/- h/- b/- b/- h h b h h

late surrender, assuming 6-7-8 loses to a naturalany 7-8 h/- h/- b/- h/- h/- b h h 9 hany 6-8 h/- h/- h/- h/- b/- b h b h hany 6-7 h/- h/- b/- h/- h/- b h h h h

KEYh IIit.

h/- See the fIrst three lines of tbe tabIe for hit or stand.s17 Dealer stands on soft seventeen.



Triple for Same SuitTriple pay for 6-7-8 of the same suit has been a rule

in Macao and a few other places in Asia. The value of thisrule to a card counter is minimal. Table 58 indicateswhen it is better to hit than to stand, when it is better tohit than to early-surrender, and when it is better to hitthan to late-surrender.


Triple for suited 6-7-8Win rate: $16Std dev: $416

6-7-8 ETC 133

Table 586-7-8 Suited Pays Triple, S17


hit/stand7-8 ss -3 -4 -5 -6 -7 h h h 9 h6-8 ss -1 -2 -3 -5 -5 h h h h h6-7 ss 2 0 0 -2 -2 h h h h h

early surrender, assuming 6-7-8 suited beats a natural7-8 ss h/- b/- h/- h/- h/- h 9 5 0 -76-8 ss h/- b/- h/- h/- h/- h h 10 3 -96-7 ss h/- b/- h/- h/- h/- h h h 7 -7

early surrender, assuming 6-7-8 suited loses to a natural7-8 ss h/- h/- h/- h/- h/- h 9 5 0 -96-8 ss h/- h/- h/- h/- h/- h h 10 2 -96-7 ss h/- h/- h/- h/- h/- h h h 7 -7

late surrender, assuming 6-7-8 suited beats a natural7-8 ss h/- b/- h/- h/- h/- h 9 5 3 76-8 ss h/- h/- h/- h/- h/- h h 10 7 h6-7 ss h/- h/- h/- h/- h/- h h h h h

late surrender, assuming 6-7-8 suited loses to a natural7-8 ss h/- h/- h/- h/- h/- h 9 5 2 46-8 ss h/- h/- h/- h/- h/- h h 10 7 h6-7 ss h/- h/- h/- h/- h/- h h h h h

KEY

b flit.

h/- See the first three lines of the table for hit or stand.

ss Same suit.




Table 59Any 6-7-8 Pays Double, HI7


hit/standany 7-8 -1 -2 -3 -4 -5 h h h h hany 6-8 1 0 -1 -2 -2 h h h h hany 6-7 5 3 1 0 0 h h h h h

early surrender, assuming 6-7-8 beats a naturalany 7-8 h/- h/- h/- h/- h/- h h 7 3 -1any 6-8 h/- h/- h/- h/- h/- h h h 8 0any 6-7 h/- h/- h/- h/- h/- h h h h 3

early surrender, assuming 6-7-8 loses to a naturalany 7-8 h/- h/- h/- h/- h/- h h 7 2 -6any 6-8 h/- h/- h/- h/- h/- h h h 6 -7any 6-7 h/- h/- h/- h/- h/- h h h h -7

late surrender, assuming 6-7-8 beats a naturalany 7-8 h/- h/- h/- h/- h/- h h 7 6 hany 6-8 h/- h/- h/- h/- h/- h h h h hany 6-7 h/- h/- h/- h/- h/- h h h h h

late surrender, assuming 6-7-8 loses to a naturalany 7-8 h/- h/- h/- h/- h/- h h 7 5 7any 6-8 h/- h/- h/- h/- h/- h h h h hany 6-7 h/- h/- h/- h/- h/- h h h h h

KEY

hh/-

h!7

number

Hit.

See the fIrst three lines of the table for hit or stand.


Stand (against 2 through 6) or surrender (against 8,9, 10, or ace) if the count per deck equals or exceedsthe number; hit if the count per deck is less than thenumber.

6-7-8 ETC 135

Table 606-7-8 Suited Pays Double, HI7


hit/stand7-8 ss -4 -5 -6 -7 -8 h h 10 7 h6-8 ss -2 -3 -4 -6 -6 h h h h h6-7 ss 1 0 -1 -3 -3 h h h h h

early surrender, assuming 6-7-8 suited beats a natural7-8 ss h/- h/- h/- h/- h/- h 8 4 0 -96-8 ss h/- h/- h/- h/- h/- h h 8 1 -96-7 ss h/- h/- h/- h/- h/- h h h 5 -7early surrender, assuming 6-7-8 suited loses to a natural7-8 ss h/- h/- h/- h/- h/- h 8 4 -1 -106-8 ss h/- h/- h/- h/- h/- h h 8 1 -96-7 ss hi- hi- hi- hi- hi- h h h 5 -7

late surrender, assuming 6-7-8 suited beats a natural7-8 ss h/- h/- h/- h/- h/- h 8 4 1 36-8 ss h/- h/- h/- h/- h/- h h 8 5 86-7 ss hi- hi- hi- hi- hi- h h h 10 hlate surrender, assuming 6-7-8 suited loses to a natural7-8 ss h/- h/- h/- h/- h/- h 8 4 1 36-8 ss h/- h/- h/- h/- h/- h h 8 5 86-7 ss hi- hi- hi- hi- hi- h h h 10 h

KEY

h IIit.

hi- See the first three lines of the tabIe for hit or stand.h 17 Dealer hits soft seventeen.

s s Same suit.



Table 61Any 6-7-8 Pays Triple, H17


hit/standany 7-8 2 0 0 -1 -1 h h h h hany 6-8 5 4 2 0 1 h h h h hany 6-7 10 7 6 4 5 h h h h h

early surrender, assuming 6-7-8 beats a naturalany 7-8 h/- h/- h/- h/- h/- h h h 7 4any 6-8 h/- h/- h/- h/- h/- h h h h 10any 6-7 h/- h/- h/- h/- h/- h h h h h

early surrender, assuming 6-7-8 loses to a naturalany 7-8 h/- h/- h/- h/- h/- h h h 6 -2any 6-8 h/- h/- h/- h/- h/- h h b h -1any 6-7 h/- h/- h/- h/- h/- h h h h 0

late surrender, assuming 6-7-8 beats a naturalany 7-8 h/- h/- h/- h/- h/- h h h h hany 6-8 h/- h/- h/- h/- h/- h h h h hany 6-7 h/- h/- h/- h/- h/- h h h h h

late surrender, assuming 6-7-8 loses to a naturalany 7-8 h/- h/- h/- h/- h/- h h h 9 hany 6-8 h/- h/- h/- h/- h/- h h h h hany 6-7 h/- h/- h/- b/- h/- h h h h h

KEY

hh/-

hI7

number

Hit.See the fIrst three lines of the table for hit or stand.


Stand (against 2 through 6) or surrender (against 8,9, 10, or ace) if the count per deck equals or exceedsthe number; hit if the count per deck is less than thenumber.

6-7-8 ETC 137

Table 626-7-8 Suited Pays Triple, HI7


hit/stand7-8 ss -3 -4 -5 -6 -7 h h h 9 h6-8 ss -1 -2 -3 -5 -5 h h h h h6-7 ss 2 0 0 -2 -2 h h h h h

early surrender, assuming 6-7-8 suited beats a natural7-8 ss h/- h/- h/- h/- h/- h 9 5 0 -76-8 ss h/- h/- h/- h/- h/- h h 10 3 -96-7 ss h/- h/- h/- h/- h/- h h h 7 -7

early surrender, assuming 6-7-8 suited loses to a natural7-8 ss h/- h/- h/- h/-- h/- h 9 5 0 -96-8 ss h/- h/- h/- h/- h/- h h 10 2 -96-7 ss hi- hi- hi- hi- hi- h h h 7 -7

late surrender, assuming 6-7-8 suited beats a natural7-8 ss h/- h/- h/- h/- h/- h 9 5 3 76-8 ss hi- hi- hi- hi- hi- h h 10 7 h6-7 ss h/- h/- h/- h/- h/- h h h h h

late surrender, assuming 6-7-8 suited loses to a natural7-8 ss h/- h/- h/- h/- h/- h 9 5 2 46-8 ss hi- hi- hi- hi- hi- h h 10 7 h6-7 ss h/- h/- h/- h/- h/- h h h h h

KEY

h lIit.

h/- See the fIrst three lines of the table for hit or stand.h 17 Dealer hits soft seventeen.

s s Same suit.



7-7-7This section applies when you get a bonus for 7-7-7.

It assumes that you are not allowed to double after splits.Table 63 gives the strategy appropriate for splitting apair of 7s when 7-7-7 (any mixture of suits) pays double,and table 64 applies when 7-7-7 pays triple.

Tables 63-64 are for use when the dealer stands onsoft seventeen; tables 65-66 are parallel and apply whenthe dealer hits soft seventeen.


Table 637-7-7 Pays Double, S17


splitany 7-7 -2 -4 -6 -7 -10 spl h h h h

early surrender, assuming 7-7-7 beats a naturalany 7-7 h/spl h/spl h/spl h/spl h/spl spl h h 7 0

early surrender, assuming 7-7-7 loses to a naturalany 7-7 h/spl h/spl h/spl h/spl h/spl spl h h 5 -4

late surrender, assuming 7-7-7 beats a naturalany 7-7 h/spl h/spl h/spl h/spl h/spl spl h h h h

late surrender, assuming 7-7-7 loses to a naturalany 7-7 h/spl h/spl h/spl h/spl h/spl spl h h 10 h

KEY

h

h/spl

spl

s17

number

Hit.

See the fIrst line of the table for hit or split.

Split.


Split (against 2 through 6) or surrender (against 8, 9,10, or ace) if the count per deck equals or exceeds thenumber; hit if the count per deck is less than thenumber.

6-7-8 ETC

Table 647-7-7 Pays Triple, S17

139


splitany 7-7 0 -I -3 -4 -7 -3 h h h h

early surrender, assuming 7-7-7 beats a naturalany 7-7 h/spl h/spl h/spl h/spl h/spl h/spl h h h 8

early surrender, assuming 7-7-7 loses to a naturalany 7-7 h/spl h/spl h/spl h/spl h/spl h/spl h h h 0

late surrender, assuming 7-7-7 beats a naturalany 7-7 h/spl h/spl h/spl h/spl h/spl h/spl h h h h

late surrender, assuming 7-7-7 loses to a naturalany 7-7 h/spl h/spl h/spl h/spl h/spl h/spl h h h h

KEYh

h/spl

sl7

number

Hit.




Double for 7-7-7Win rate: $16Std dey: $415

Triple for 7-7-7Win rate: $16Std dey: $416


Table 657-7-7 Pays Double, HI7

Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 Asplitany 7-7 -2 -4 -6 -7 -10 spl h h h h

early surrender, assuming 7-7-7 beats a naturalany 7-7 h/spl h/spl h/spl h/spl h/spl spl h h 7 0

early surrender, assuming 7-7-7 loses to a naturalany 7-7 h/spl h/spl h/spl h/spl h/spl spl h h 5 -4

late surrender, assuming 7-7-7 beats a naturalany 7-7 h/spl h/spl h/spl h/spl h/spl spl h h h h

late surrender, assuming 7-7-7 loses to a naturalany 7-7 h/spl h/spl h/spl h/spl h/spl spl h h 10 h

KEY

h

h/spl

hI7

spl

number

Hit.



Split.


6-7-8 ETC

Table 667-7-7 Pays Triple, Ht7

141


splitany 7-7 0 -1 -3 -4 -7 -3 h h h h

early surrender, assuming 7-7-7 beats a naturalany 7-7 h/spl h/spl h/spl h/spl h/spl h/spl h h h 8

early surrender, assuming 7-7-7 loses to a naturalany 7-7 h/spl h/spl h/spl h/spl h/spl h/spl h h h 0

late surrender, assuming 7-7-7 beats a naturalany 7-7 h/spl h/spl h/spl h/spl h/spl h/spl h h h h

late surrender, assuming 7-7-7 loses to a naturalany 7-7 h/spl h/spl h/spl h/spl h/spl h/spl h h h h

KEY

h

h/spl

h17

number

Hit.

See the frrst line of the table for hit or split.




CHAPTER 11

OVER/UNDER 13

The over/under is two side bets. You can bet thatyour first two cards will total over thirteen, or that theywill total under thirteen. Aces count one, and thirteenloses.

This innovation started in 1988 at Caesars Tahoeand hopefully will continue to spread.

Basic StrategyBasic strategy says ignore the over/under side

bets. You can make money on them only by countingcards.

Both the over and under are horrible bets rightafter a shuflle. The over loses at the rate of more than 6%and the under loses at a rate of more than 10%.

OVER/UNDER 143

Single DeckWhat makes the over/under beatable is volatility.

As a card is dealt out and set aside not to be reused untilafter the next shufile, the change in the casino's edge ismuch greater for the over/under bets than for regularblackjack.

Here is an example showing how strong the overbet is. With a freshly-shuffled single deck, there are 1326two-card combinations, of which 618 exceed thirteenand win while 708 lose, for a casino edge of 6.8%.Removing two small cards leaves 50 cards from whichcan be dealt 1225 two-card combinations, of which 618win and 607 lose, for a player edge of 0.9%. Removing athird small card leaves 49 cards from which can be dealt1176 two-card combinations, of which 618 win and 558lose, for a player edge of 5.1%. Removing a fourth smallcard leaves 48 cards from which can be dealt 1128 twocard combinations, of which 618 win and 510 lose, for aplayer edge of 9.6%. Removal of one card per deck canmake a difference of about 4% in expected value of theover bet. The most effect any card has in regular blackjack is about 0.8% for removal of a 5.

You do not fmd profitable over or under bets veryoften, but some of the bets you make have very highexpected values. When the Comstock in Reno firstintroduced the over/under bets at single and doubledeck, it had a $100 maximum on those bets and wascasual about $100 bets. I made mostly flat $100 betsand played basic strategy, and when the crush count(explained later in this chapter) told me I had an edge onthe over or the under, I made the appropriate side betwith another $100 chip. One time after I placed my betson two hands, each with an over bet of $100, but before


the dealer started to distribute cards, the action washalted for replenishing the dealer's chip supply. Whilewaiting, I calculated my expected win on the cominghand. I had a running crush count of +9 with about halfa deck remaining to be dealt. At 2% per count per deck,my edge was 30% on my over bets! A shufile would havecost me $60 in expected value! Mentally I shouted"Please don't shuffle! Please don't shuffle!" Fortunatelythe dealer did not shuffle; she dealt me two hands overthirteen.

The single-deck over/under simulation results reported here are for benchmark rules plus over/under. Atthe Comstock during its big over/under giveaway, dealers hit soft seventeen and players could double downonly on ten or eleven. Those two rules variations reducedthe win rate to $62.

For single or double deck I suggest playing basicstrategy and not deviating from it based on the count,except to make profitable over/under bets.

Over/under, high-low,one deck


Over/under, crush count,one deck


OVER/UNDER 145

You will note from the simulation results that thestandard deviation is higher when you occasionally betthe over or under. This increased risk means you maywant to decrease your bet size somewhat or take a largerbankroll to handle the larger fluctuations you willencounter.

Counting CardsCommonly-used counting systems are not well

suited to the over/under decision. For the high-low, betthe under at counts below -6 and bet the over at countsabove 5. Those numbers apply to all numbers of decks.Counts that far from zero do not occur often, reflectingthe fact that the high-low is not a good counting systemto use for the over/under.

Counting systems that are good for the over arealso good for the under.

The pioneering publication explaining how to getan edge on the over/under side bets is "The Over/UnderReport," an excellent paper by Arnold Snyder. I havechecked the numbers in this paper, including calculating all the correlation coefficients for one to eight decks,and I found no mistakes.

I am not going to reproduce Snyder's work here.But I will introduce the counting system that I prefer touse for the over/under side bets, a system not mentionedin Snyder's paper. It was called the crush count by JakeSmallwood in the March 1990 issue of Blac/ifack Monthly.The crush count has a correlation coefficient of 0.97 for


Table 67Crush Count

Card CountAce,2 +2

3 +14,5,6,7,8 0

9,10,J,Q,K -1

the over and -0.95 for the under. It is a level 2 count,meaning that the values you assign to the cards areintegers from -2 to +2. Snyder's best level 2 count isslightly more efficient at 0.98 for the over and -0.96 forthe Wlder. I prefer the crush COWlt because it assigns thevalue of zero to more cards, making it easier to use.

The crush count is shown in table 67. It counts +2for A and 2, +1 for 3, and -1 for 9-10. Cards 4-5-6-7-8 count

Over/under, six decks,high-low COWlt


Over/under, six decks,crush count


Over/under, six decks,crush for over/under onlyhigh-low for other decisions


OVER/UNDER 147

zero. Bet the over on crush counts per deck of 3 or more,and bet the under on crush counts per deck of -5 or lower.Those numbers apply to all numbers of decks. (A reminder: -6 is lower than -5; -4 is not lower than -5.)

Each count per deck of the crush count changesyour expectation on over and under bets by about 2.1%.The break-even count per deck on the over is 3.1 forsingle and double deck, and 3.0 for more than two decks.The break-even count per deck on the under is -4.6 forall numbers of decks.

You can get a healthy edge at the over/under, if youcan play without the dealer's shuftling up on you. Thesimulation results reported in the rectangles are for thebenchmark betting scheme, including a matching bet onthe over or under if justified. Better penetration meansa higher expected win rate, probably more so than withany other blackjack rule.


CHAPTER 12

HALVES COUNT

The counting system introduced in chapter 3 iscalled the high-low system. The counting system introduced in this section will be referred to as countinghalves. Both counting systems have the same efficiencywith respect to insurance and each requires the sameamount of memorizing from tables. Counting halves ismore difficult to do, but will result in a slightly higherexpected-win rate. Counting halves means assigningvalues to cards as shown in table 68. (If you prefer towork with integers you can double all these card values,and also double the strategy numbers in tables 69 and70.)

A full deck contains 20 plus and 20 minus points inthe high-low system, and 22 plus and 22 minus points inthe halves system. The halves system finds more situations of advantage than the high-low system does. The

HALVES COUNT 149

halves count is a very close approximation of the precisevalue of each card as given on page 162 of the 1962edition of Thorp's Beat the Dealer. With typical rules, thedealer's advantage disappears when either high-low orhalves count per deck rises to +1. According to Thorp'sbook, each additional count per deck gives you an extra0.56% of advantage in the high-low count, with a standard deviation of 0.22%. Thus the value of a high-lowcount per deck plus or minus one standard deviation is0.34% to 0.78%. (Chapter 14 will explain standarddeviation.) In the halves count, each additional countper deck also gives you an extra 0.56% advantage

HalvesWin rate: $17Std dev: $426

Table 68Comparison of High-Low and

Halves CountsCard

23456789

10,J,Q,Kace

High-Low

11111ooo

-1-1

Halves

0.51.01.01.51.00.50.0

- 0.5-1.0-1.0


according to Thorp's book, but with a standard deviationof only 0.06%. These numbers come from comparisons ofthe counting systems with Thorp's card values. The lowstandard deviation means that the halves count makesalmost no errors in estimating advantage.

Peter Griffin in The Theory Of Blackjack says thebetting correlation of the halves counting system is 0.99and the playing correlation is 0.58.

Simulation shows the halves count results in higherw.in per hour by $1 for the benchmark rules, whencompared to the high-low. The sample sizes were increased to the point where that difference is statisticallysignificant.

The high-low count overestimates the values of 2and 9 and underestimates the values of 5 and 7. Thehalves count is a very close approximation of the truevalue of each card. As an example, suppose you arewatching two tables simultaneously, both tables aresingle deck, and both dealers are shuffling. One dealeris a bit careless and shows you the bottom card, a 5. Theother dealer is even more careless and shows you boththe burned card and the bottom card, and they are both2s. You should immediately make a bet at one tablebecause you have an advantage, but which table? According to Thorp's data, one 5 gone gives you a greateradvantage than two 2s gone. The halves count tells youaccurately that the table with the 5 gone is better, sinceit gives you counts of +1.5 and +1. The high-low countgives you the wrong answer in this case, since it givesyou counts of +1 and +2. (Griffin claims that a 5 is worthalmost exactly the same as two 2s.)

An attractive feature of counting halves is that thestrategy tables are similar to those of the high-lowsystem. Once you are expert with the high-low system,

HALVES COUNT 151

you can switch to counting halves in easy stages. Forexample, you can count halves but make your playingstrategy decisions by using the high-low tables that youhave already memorized; and, if counting halves is toodifficult at fIrst you can always switch back to the highlow system. When you feel so confident counting halvesthat you know you do not want to go back to the high-lowsystem, then learn the halves strategy numbers andforget the high-low strategy numbers.

Table 69 is the halves strategy numbers for dealerstands on soft seventeen; it is the counterpart of table 9(page 46). Table 70 is the halves strategy numbers fordealer hits soft seventeen; it is the counterpart to table10 (page 48).

InsuranceInsurance is a good bet if the halves count per deck

exceeds +1.7 in a single-deck game, +2.7 in a doubledeck game, +3.2 with four decks, and +3.3 with six decks.

Over/Under 13The over is a good bet at counts per deck of +5 and

higher. The under is a good bet at counts per deck of -6and lower.

Appendix BAppendix B contains strategy rules for the halves

system. Table B1 is for a single deck when the dealer hitssoft seventeen. Table B2 is for four decks when the dealerhits soft seventeen. Table B3 is for a single deck when thedealer stands on soft seventeen. Table B4 is for fourdecks when the dealer stands on soft seventeen.



Table 69Halves Strategy Numbers, S17


double not allowed after splitA-A spl spl spl spl spl -10 -9 -8 -8 -410-10 8 6 5 59-9 -1 -2 -3 -4 -4 5 -9 -10 38-8 spl spl spl spl spl spl spl spl 5* spl7-7 -8 -8 -10 spl spl spl h h h h6-6 1 0 -3 -5 -7 h h h h h5-5 never split - play as total of ten4-4 never split - play as total of eight3-3 9 3 0 -2 spl 8* h h h h2-2 6 3 0 -3 -8 spl h h h h

ace-9 10 8 6 5 4ace-8 7 5 3 1 1ace-7 -/db 0 -1 -4 -7 -8 h h h/-ace-7 bI- h h 1ace-6 1 -2 -5 -9 db h h h h hace-5 h 3 -2 -6 db h h h h hace-4 h 6 0 -5 -10 h h h h hace-3 h 7 1 -1 -5 h h h h hace-2 h 7 3 0 -2 h h h h h

hard 20hard 19hard 18hard 17 -6hard 16 -9 -10 9 8 5 0 8hard 15 -5 -6 -8 -9 -10 10 h 9 4 9hard 14 -3 -4 -5 -6 -7 h h h 8 hhard 13 0 -1 -3 -4 -4 h h h h hhard 12 3 2 0 -1 0 h h h h h11 db db db db db -9 -7 -4 -4 110 -8 -9 -10 db db -6 -4 -1 4 49 0 0 -2 -4 -6 3 8 h h h8** h 8 5 3 1 h h h h h7 h h h 9 10 h h h h h6 h h h h h h h h h h5 h h h h h h h h h h

HALVES COUNT 153


double allowed after splitA-A spl spl spl spl spl -10 -9 -8 -9 -410-10 8 6 5 59-9 -2 -3 -4 -6 -6 3 -9 -10 38-8 spl spl spl spl spl spl spl spl 8* spl7-7 -9 -9 spl spl spl spl 2 h h h6-6 -2 -4 -6 -8 spl h h h h h5-5 never split - play as total of ten4-4** h 7 1 -2 -5 h h h h h3-3 0 -5 -8 -10 spl spl h h h h2-2 -2 -4 -6 -8 -10 spl 4 h h h

KEY

-/db

db

h

h/-

spl

s17

number

*

**

Stand.


Double down.

Hit.


Split.






Table 70Halves Strategy Numbers, H17


double not allowed after splitA-A spl spl spl spl spl -10 -9 -8 -9 -510-10 8 6 5 49-9 -1 -2 -3 -4 -6 5 -9 -10 28-8 spl spl spl spl spl spl spl spl 5* -17-7 -7 -8 -10 spl spl spl h h h h6-6 1 0 -3 -5 -8 h h h h h5-5 never split - play as total of ten4-4 never split - play as total of eight3-3 8 3 0 -2 -6 8* h h h h2-2 6 2 0 -3 -6 spl h h h h

ace-9 10 8 6 5 4ace-8 7 4 3 1 0ace-7 -/db 0 -1 -5 -7 db h h hace-7 hI- h h hace-6 1 -2 -5 -9 db h h h h hace-5 h 3 -2 -6 db h h h h hace-4 h 5 0 -5 db h h h h hace-3 h 6 1 -2 -6 h h h h hace-2 h 7 3 0 -3 h h h h h

hard 20hard 19hard 18hard 17 -4hard 16 -9 -10 9 8 5 0 3hard 15 -6 -7 -8 -9 10 h 9 4 5hard 14 -3 -4 -5 -7 -9 h h h 10 8hard 13 0 -1 -3 -4 -6 h h h h hhard 12 3 1 0 -1 -3 h h h h h

11 db db db db db -9 -7 -4 -4 010 -9 -9 -10 db db -6 -4 -1 4 39 1 0 -2 -4 -6 3 8 h h h8** h 8 5 3 1 h h h h h7 h h h 9 9 h h h h h6 h h h h h h h h h h5 h h h h h h h h h h

HALVES COUNT 155


double allowed after splitA-A spl spl spl spl spl -10 -9 -8 -9 -510-10 8 6 5 49-9 -3 -4 -5 -6 -8 3 -9 -10 18-8 spl spl spl spl spl spl spl spl 8* -17-7 spl spl spl spl spl spl 2 h h h6-6 -2 -4 -6 -8 spl h h h h h5-5 never split - play as total of ten4-4** h 6 1 -2 -7 h h h h h3-3 0 -4 -7 -10 spl spl h h h h2-2 -2 -4 -6 -8 -10 spl 4 h h h

KEY

-Idb

db

hh17

h/-

spl

number

*

**

Stand.


Double down.

Hit.


Do not double. See the hI- row for advice on whetherto hit or to stand.

Split.





CHAPTER 13DOUBLE

EXPOSURE

In double exposure, both of the dealer's cards areface up for you to see before you play your hand. Thissounds great, but there is another rule that is to yourdisadvantage: The dealer wins pushes unless you bothget naturals. At most casinos with double exposure,naturals win even money instead of three to two butyour natural wins even if the dealer also has a natural.

Blackjack Secrets has a chapter on double exposure.That book has basic strategy and high-low strategynumbers from -1 to +6. A few paragraphs of this bookare word-for-word copies from that chapter of BlackjackSecrets. This book has three things the Blackjack Secretschapter does not have: high-low strategy numbers for a

DOUBLE EXPOSURE 157

broader range (-10 to +10), strategy numbers for thehalves counting system, and simulation results.

Estimating Your EdgeSimulation shows a basic-strategy player's approxi

mate expectation for double exposure for four or moredecks, dealer stands on soft seventeen, no resplits, andnaturals pay even money, is -0.2%. (The negative signmeans the casino has the edge.) This is shown in table71. The rest of table 71 contains numbers that you canadd to that -0.2%, depending on how the rules you facevary from those benchmark rules. For example, if youare playing a double-exposure game in which the dealer

Table 71Player Edge for Dbl Exposure

benchmark (both dealer cards face up,pushes lose except natural against natural,dealer stands on soft seventeen,double any first two cards, no resplits) -0.2%

to the above add the following:rule change in player edge

dealer hits soft seventeen -0.4%naturals pay 3:2 2.3tied naturals win 0.2resplits allowed 0.3double after splits 0.2no split unlike 10s -0.2double ten & eleven only -1.3double nine, ten & eleven only -0.9


hits soft seventeen, the effect on your expectation is-0.4%, bringing your total expectation to -0.6%.

If you get paid though your natural is tied by thedealer's natural, table 71 says you gain 0.2%. If naturalspay 3:2, the value of getting paid on a tied natural is0.3%.

Table 71 shows that allowing double after splitsgives the player 0.3%. Of that, 0.1% comes from doublingon less than nine. So if you can double after splits butonly on totals of nine or more, you gain 0.2% rather than0.3%.

Basic StrategyTables 72-74 contain generic basic strategy for

double exposure.Table 72 contains the part of basic strategy for

double exposure that applies no matter what the dealerdoes on soft seventeen. In this table, split strategy is thesame whether or not you can double down after splitting.

Table 73 is the rest of basic strategy for use if thedealer stands on soft seventeen. Table 74 is the rest ofbasic strategy for use if the dealer hits soft seventeen.Thus a complete basic strategy for double exposure istable 72 plus either table 73 or table 74.

There are important differences in strategy thatdepend on whether the dealer hits or stands on softseventeen. One important difference is if the dealerstands on soft seventeen, then A-6 is played according tothe column labeled 17 in table 72. If the dealer hits softseventeen, there is a separate column for that hand (intable 74).

With soft eighteen against small dealer totals, itdoes not matter much whether you hit or stand if you

DOUBLE EXPOSURE 159

cannot double. Eighteen is not a strong hand when thedealer wins ties.

You might be wondering why basic strategy saysstand on sixteen against eight at double exposure, buthit at single exposure. Standing on sixteen has equalvalue at either variety of blackjack - you win if thedealer busts, and you cannot push. Hitting sixteen ismore valuable at single exposure because you might endup with a push. The dealer wins ties at double exposure.

Seventeen is the worst hand you can get at doubleexposure - you cannot push, and the only way you canwin with seventeen is if the dealer busts.

Counting CardsEach count per deck is worth about 0.7% at double

exposure (see page 290). This is in contrast to the 0.5%per count per deck at single-exposure blackjack. Youradvantage at double exposure is more volatile than youradvantage at single exposure - at higher counts youradvantage is higher. Therefore, for equivalent bet sizesyou can make more money at double exposure than atsingle exposure.

Counting Cards: High-LowThe high-low counting system, used in this book for

single-exposure blackjack, is also the best simple counting system for double exposure. This chapter presentsdouble-exposure strategy numbers for the high-lowcount.

Table 75 provides the strategy numbers that applyno matter what the dealer does on soft seventeen. Table76 is for use if the dealer stands on soft seventeen. Table77 is for use if the dealer hits soft seventeen. All thecalculations in tables 75-77 are based on six decks.



Table 72Double Exposure Basic, Part 1

Player's Dealer's TotalHand 7 8 9 10 11 12 13 14 15 16 17 18 19 20

double not allowed after splitA-A spl spl spl spl h spl spl spl spl spl h h h h10-10 - spl spl spl spl - h9-9 - spl - - spl spl spl spI spl - spl h h8-8 spl spl - - spl spI spl spI spl spl h h h7-7 h h h h - spl spl spl spl spl spl h h h6-6 h h h h h spl spl spl spl spl spl h h h5-5 db db h h h db db db db db h h h h4-4 h h h h h spl spl spl spl spl h h h h3-3 h h h h h spl spl spl spl spl spl h h h2-2 h h h h h spl spl spl spl spl spl h h h

soft 21soft 20 - dbs dbs dbs dbs - hsoft 19 - dbs dbs dbs dbs dbs - h hsoft 18 h h h h dbs db db db db - h h hsoft 17 h h h h h db db db db db h h h hsoft 16 h h h h h db db db db db h h h hsoft 15 h h h h h db db db db db h h h hsoft 14 h h h h h db db db db db h h h hsoft 13 h h h h h db db db db db h h h h

hard 21 -hard 20 - hhard 19 - h hhard 18 - h h hhard 17 - h h h hhard 16 h h h h hhard 15 h h h h h h hhard 14 h h h h h h h hhard 13 h h h h h h h h hhard 12 h h h h h h h h h

11 db db db h h db db db db db h h h h10 db db h h h db db db db db h h h h9 h h h h h db db db db db h h h h8 h h h h h db db db db db h h h h7 h h h h h h h db db db h h h h6 h h h h h h h db db db h h h h5 h h h h h h h db db db h h h h

DOUBLE EXPOSURE 161

Table 72 ContinuedPlayer's Dealer's TotalHand 7 8 9 JO JJ J2 J3 J4 15 J6 J7 J8 J9 20

double allowed after splitA-A spl spl spl spl h spl spl spl spl spl h h h h10-10 - spl spl spl spl - h9-9 - spl - - spl spl spl spl spl - spl h h8-8 spl spl - - spl spl spl spl spl spl h h h7-7 h h h h h spl spl spl spl spl spl h h h6-6 h h h h h spl spl spl spl spl spl h h h5-5 db db h h h db db db db spl h h h h4-4 h h h h h spl spl spl spl spl h h h h3-3 h h h h h spl spl spl spl spl spl h h h2-2 h h h h h spl spl spl spl spl spl h h h

KEY

Stand.


dbs Double down; if you cannot double, then stand.

h IIit.

spl Split.


Table 73Double Exposure Basic, S17

Player's Dealer's TotalHand 4 5 6 A-A A-2 A-3 A-4 A-5

double not allowed after splitA-A spl spl spl spl spl spl spl spl10-109-9 spl spl spl8-8 spl spl spl spl spl7-7 spl spl h6-6 spl spl spl h h h h5-5 db db db h h db db db4-4 h h h h h h h h3-3 h h spl h h h h h2-2 h h spl h h h h h

soft 21soft 20soft 19soft 18 dbs dbs dbs h h h hsoft 17 h db db h h h h hsoft 16 h db db h h h h hsoft 15 h h db h h h h hsoft 14 h h h h h h h hsoft 13 h h h h h h h h

hard 21hard 20hard 19hard 18hard 17hard 16hard 15hard 14 hhard 13 h hhard 12 h h h h

11 db db db h db db db db10 db db db h h db db db9 h db db h h h h h8 h h h h h h h h7 h h h h h h h h6 h h h h h h h h5 h h h h h h h h

163DOUBLE EXPOSURE

Table 73 ContinuedPlayer's Dealer's TotalHand 4 5 6 A-A A-2 A-3 A-4 A-5

double allowed after splitA-A spl spl spl spl spl spl spl spl10-109-9 spl spl spl spl8-8 spl spl spl spl spl spl7-7 spl spl spl h6-6 spl spl spl h h h h5-5 db db db h h db db db4-4 h h h h h h h h3-3 spl spl spl h h h h h2-2 spl spl spl h h h h h

KEY

Stand.



h IIit.

spl Split.



Table 74Double Exposure Basic, HI7

Player's Dealer's TotalHand 4 5 6 A-A A-2 A-3 A-4 A-5 A-6

double not allowed after splitA-A spl spl spl spl spl spl spl spl spl10-109-9 spl spl spl spl spl8-8 spl spl spl spl spl7-7 spl spl h6-6 spl spl spl h h h5-5 db db db h h db db db h4-4 h h h h h h h h h3-3 h h spl h h h h h h2-2 h h spl h h h h h h

soft 21soft 20soft 19soft 18 dbs dbs dbs h h h h hsoft 17 db db db h h h h h hsoft 16 h db db h h h h h hsoft 15 h h db h h h h h hsoft 14 h h db h h h h h hsoft 13 h h h h h h h h hhard 21hard 20hard 19hard 18hard 17 hhard 16 hhard 15 hhard 14 hhard 13 hhard 12 h h h1] db db db h db db db db h10 db db db h h db db db h9 h db db h h h h h h8 h h h h h h h h h7 h h h h h h h h h6 h h h h h h h h h5 h h h h h h h h h

165DOUBLE EXPOSURE

Table 74 ContinuedPlayer's Dealer's TotalHand 4 5 6 A-A A-2 A-3 A-4 A-5 A-6

double allowed after splitA-A spl spl spl spl spl spl spl spl spl10-109-9 spl spl spl spl spl8-8 spl spl spl spl spl spl7-7 spl spl spl h6-6 spl spl spl h h h5-5 db db db h h db db db h4-4 h h spl h h h h h h3-3 spl spl spl h h h h h h2-2 spl spl spl h h h h h h

KEY

Stand.



h IIit.


spl Split.


Blackjack Count Analyzer analyzes the many variationsof double exposure, and was the tool used to develop thestrategy numbers in this book.

For decision purposes you need a count per deck.The method for finding it is the same as was discussedin chapter 3. Start with zero after a shuffle. Keep arunning count, counting all cards you have seen sincethe shuffle. Divide this running count by the number ofdecks you have not seen to get the count per deck. Youshould stand, hit, or double down only if your count perdeck (truncated, and not rounded) is equal to or greaterthan the number in the table.

Insurance. Insurance at double exposure is thesame as is explained in chapter 4, and the strategynumbers of table 11 (page 53) are applicable to doubleexposure too. Generally double exposure uses six decks,which means an insurance strategy number of 3.0.

What the numbers mean. You might want todouble down on a natural against fourteen, fifteen, orsixteen if the count is high enough. This is true only ifnaturals pay only even money. Do not double down on anatural if it pays 3:2.

Suppose you have 4-4 and have a choice of hitting,splitting, or doubling down. Regardless of the dealer'saction on soft seventeen, meaning you are using eithertable 76 or 77, doubling down on your total of eight ismore profitable than splitting 4-4 if the dealer has six orless, or soft seventeen or less. If the dealer has hardtwelve to sixteen, then basic strategy says splitting 4-4is more profitable than doubling down on eight. Whenthe count per deck gets above the numbers shown intable 75, then doubling on eight is more profitable thansplitting 4-4.

DOUBLE EXPOSURE 167

Against twelve or thirteen, hit 4-4 if the count perdeck is bad enough. The count per deck must be at least-2 to justify splitting 4-4 against twelve, and it must beat least -7 to justify splitting 4-4 against thirteen.

With 6-6 against seventeen, split if the count perdeck is less than +2 and hit if the count per deck is +2or more.

With 3-3 and 2-2, splitting is always more profitable than doubling down.

The numbers for soft twenty against twelve andthirteen are stand/double. If the count per deck is equalto or greater than the number in the table, then doubledown.

With soft eighteen against twelve through sixteen,double if you can. The table numbers for soft eighteenagainst twelve through sixteen are hit/stand if youcannot double. For example, if you have soft eighteenand the dealer shows thirteen, double if you can. If youcannot double, then hit if the count per deck is lessthan zero or stand if the count per deck is zero or more.

The numbers for soft eighteen against dealer A-Athrough A-5 are for when to double.

When the dealer stands on soft seventeen, youhave soft eighteen, and the dealer shows A-A throughA-5, if you do not double it does not matter whether youhit or stand. So take a card on a negative count tohasten the shuftle, and stand on a positive count.

With 7-7, if you do not split then either hit or standaccording to the strategy number for fourteen. Likewise with 6-6: If you do not split, then play according tothe strategy number for twelve.



Table 75High-Low, Dbl Exposure, Part 1

Player's Dealer's TotalHand 7 8 9 JO JJ J2 J3 J4 15 16 17 J8 J9 20double not a/lowed after splitA-A -7 -6 -6 -4 h spl spl spl spl spl 4 7 8 710-10 2 0 -2 -4 -9 h9-9 9 spl - - -8 spl spl spl spl - spl h h8-8 spl spl h/- h/- h/- spl spl spl spl spl spl -3* h h7-7 h/- h/- h/- h/- h/- -9 spl spl spl spl spl h h h6-6 hI- h/- hI-hi- hI- - lOspIspispispI 2* h h h5-5 never split - playas total of ten4-4 h h h h h -2/5 -7/4 4 4 2 -9* -8* h h3-3 h h h h h -6 -10 spl spl spl spl-l0* h h2-2 h h h h h -6 -10 spl spl spl spl h h h

soft 21 - 10 6 3 1soft 20 2 0 -2 -4 -9 - hsoft 19 - -2 -5 -8 -10 dbs - h hsoft 18 h h h h 0 0 0 0 0 h h hsoft 17 h h h h h db db db db db h h h hsoft 16 h h h h h db db db db db h h h hsoft 15 h h h h h -10 db db db db h h h hsoft 14 h h h h h -6 db db db db h h h hsoft 13 h h h h h -4 -8 db db db h h h h

hard 21 -hard 20 - hhard 19 - h hhard 18 - h h hhard 17 -7 -6 -8 h h h hhard 16 0 0 -2 -6 -8 - h h h hhard 15 5 4 I -2 -4 - h h h hhard 14 h h 10 2 -2 - h h h hhard 13 h h h h 1 h h h hhard 12 h h h h 7 -8 -10 - h h h h

11 -6 -4 -1 h h db db db db db 10 h h h10 -4 -1 3 h h db db db db db 8 h h h9 6 h h h h -7 -10 db db db h h h h8 h h h h h 0 -3 -7 -9 db h h h h7 h h h h h 5 0 -2 -6 db h h h h6 h h h h h 7 2 -1 -4 db h h h h5 h h h h h 7 3 0 -4 db h h h h4 h h h h h 8 3 0 -3 db h h h h

DOUBLE EXPOSURE 169

Table 75 ContinuedPlayer's Dealer's TotalHand 7 8 9 JO Jl 12 J3 14 15 J6 J7 J8 J9 20

double allowed after split

A-A -7 -6 -6 -4 h spl spl spl spl spl 4 7 8 710-10 2 0 -2 -4 -9 h9-9 7 spl - -9 spl spl spl spl - spl h h8-8 spl spl h/- h/- h/- spl spl spl spl spl spl -3* h h7-7 8 h/- h/- h/- h/- -9 spl spl spI spI spl h h h6-6 hi- hi- hi-hi- hi- -lOspIspIspisp I 2* h h h5-5 never split - playas total of ten4-4 h h h h h -6 spl spl spI spl -9* -8* h h3-3 8 h h h h -9 spl spl spl spl spl-IO* h h2-2 5 h h h h -9 spl spl spl spl spl h h h

KEY

Stand.


Double down; if you cannot double, then stand.

Hit.

Hit or double; never split.

Hit or stand. Never split.

Split.


Split if the count per deck is less than the number;else hit.

numbers see text.

*

two

db

dbs

h

h/db

h/-

spl

number

Dbl exp benchmarkWin rate: $27Std dey: $437


Table 76High-Low, Dbl Exposure, 817


double not allowed after splitA-A spl spl spl -3 -4 -5 -6 -810-10 8 6 6 10 99-9 -2 -3 -3 9 6 4 2 08-8 spl spl spl 7 3 1 -1 -27-7 1 -1 -5 h/- h/- h/- 7 56-6 0 -3 -5 h/- h/- h/- 6 45-5 never split - play as total of ten4-4 never split - play as total of eight3-3 3 0 -1 h h h 8 72-2 3 0 -1 h h h 8 6

soft 21soft 20 8 6 6 10 9soft 19 6 3 3 8 7soft 18 -2 -5 -7 h h 9 4 2soft 17 3 -1 -6 h h h 9 7soft 16 5 0 -5 h h h 10 9soft 15 6 1 -1 h h h 10 10soft 14 6 2 0 h h h 10 9soft 13 6 3 2 h h h 10 9

hard 21hard 20hard 19hard 18hard 17 -5 -6 -8 -10hard 16 -3 -4 -6 -7hard 15 -10 -1 -2 -4 -5 -9hard 14 -9 -10 0 0 -2 -4 -7hard 13 -6 -8 -8 3 1 0 -2 -5hard 12 -2 -4 -4 6 4 2 0 -1

11 db db db 0 -1 -2 -4 -610 -8 -9 db 4 2 0 -1 -39 1 0 -3 h h 8 5 48 9 6 4 h h h h 107 h h h h h h h h6 h h h h h h h h5 h h h h h h h h

DOUBLE EXPOSURE


double allowed after splitA-A spl spl spl -3 -4 -5 -6 -810-10 8 6 6 10 99-9 -3 -5 -5 8 6 3 1 08-8 spl spl spl 5 2 0 -2 -37-7 -2 -4 -8 b/- h/- 8 4 26-6 -3 -5 -7 b/- h/- 8 4 15-5 never split - play as total of ten4-4 5 1 0 h h h 9 93-3 0 -2 -5 h h 8 5 32-2 0 -2 -5 h h 8 5 3

KEY

171

db

h

h/-

spl

s17

number

Stand.


Hit.


Split.




Table 77High-Low, Db) Exposure, H17


double not allowed after splitA-A spl spl spl -4 -5 -6 -6 -8 -310-10 8 6 5 9 99-9 -2 -4 -6 7 4 1 0 -18-8 spl spl spl h/- 7 4 1 0 -97-7 1 -1 -4 h/- h/- 10 6 4 h6-6 0 -3 -6 h/- h/- 10 6 4 h5-5 never split - play as total of ten4-4 never split - play as total of eight3-3 3 0 -1 h h h 7 6 h2-2 3 0 -2 h h 10 7 5 h

soft 21soft 20 8 6 5 9 9soft 19 5 3 1 10 7 6soft 18 -3 -5 -9 h h 9 5 3soft 17 3 -1 -6 h h h 8 6 hsoft 16 4 0 -5 h h h 9 7 hsoft 15 5 1 -2 h h h 9 8 hsoft 14 6 2 0 h h h 9 8 hsoft 13 6 3 0 h h h 9 8 h

hard 21hard 20hard 19hard 18hard 17 -9 -10 3hard 16 -6 -7 -9 -10 6hard 15 -4 -5 -6 -8 8hard 14 -9 -10 -2 -3 -5 -7 -9 hhard 13 -7 -8 0 -1 -3 -4 -7 hhard 12 -3 -4 -7 3 1 0 -1 -3 h

11 db db db 0 -1 -3 -4 -6 010 -8 -9 db 4 2 0 -1 -3 29 1 0 -3 h h 8 5 4 h8 9 6 4 h h h h 10 h7 h h h h h h h h h6 h h h h h h h h h5 h h h h h h h h h

DOUBLE EXPOSURE 173


double allowed after splitA-A spl spl spl -4 -5 -6 -6 -8 -310-10 8 6 5 9 99-9 -3 -5 -7 6 3 1 0 -28-8 spl spl spl 9 5 2 0 -1 -97-7 -2 -4 -8 h/- h/- 8 4 2 h6-6 -3 -5 -8 h/- h/- 7 3 1 h5-5 never split - play as total of ten4-4 4 1 -1 h h h 7 6 h3-3 0 -2 -5 h h 8 5 3 h2-2 0 -2 -5 h h 7 4 2 h

KEY

db

h

h17

h/-

spl

number

Stand.


Hit.



Split.



Counting Cards: Hi-OptThe hi-opt is another simple counting system. In it,

3, 4, 5, 6 count +1, lOis -1, and the· other cards countzero. Earlier editions of Professional Blackjack presented strategy numbers for double exposure for hiopt. That feature has been removed from this bookbecause simulation of it as a simple count (i.e. nosecond count of aces) gave a win rate of $26, which islower than that of the high-low.

Counting Cards: HalvesSimulations show that the halves counting system

handily outperforms the high-low at double exposure.Moving up to the halves count for deciding when to betbig means a bigger increase in expected win rate thanlearning strategy numbers.

For decision purposes you need a count per deck.The method for finding it is the same as was discussedin chapter 3. Start with zero after a shuffle. Keep arunning count, counting all cards you have seen sincethe shuft1e. Divide this running count by the number ofdecks you have not seen to get the count per deck.

Insurance for the halves count at double exposureis the same as is explained in chapter 4, and thestrategy numbers of table 11 (page 53) are applicableto double exposure too. Generally double exposure usessix decks, which means an insurance strategy numberof 3.3.

For the halves count, use table 78 with either 79 or80. Use tables 78 and 79 if the dealer stands on softseventeen. Use tables 78 and 80 if the dealer hits softseventeen. The calculations for tables 78-80 are basedon six decks.

Hit or stand decisions with soft eighteen againstfour to six or twelve to sixteen are insensitive to the

DOUBLE EXPOSURE 175

halves count. For all of these dealer hands basic strategy is neutral - it does not matter whether you hit orstand whatever the count.

Win Rates at Double ExposureTable 81 contains simulation results for double

exposure. The format is the same as for the rest of thewin rates in this book. The first row is benchmark winrates, and the rest of the rows are win rates that differfrom the benchmark in the identified manner.

The benchmark is six decks, five dealt out; ties lose;dealers stand on soft seventeen; player can double anyfirst two cards but not double after splitting; no resplits;naturals pay even money, and tied naturals push. Thebenchmark rules are used in simulations of the highlow and the halves counts.

For simulations of deviations from the benchmarks,results are presented both as raw expected-win ratesand as deviations from the appropriate benchmarkwin rates.

Note that varying bets with the count wins almostas fast when you use basic strategy as when you varyyour strategy with the count. That means all the strategy numbers of this chapter put together have only amodest effect on your win rate.



Table 78Halves, Dbl Exposure, Part 1

Player's Dealer's TotalHand 7 8 9 10 11 12 13 14 15 16 17 18 19 20

double not allowed after splitA-A -7 -6 6 -5 h spl spl spl spl spl 5 6 8 910-10 2 0 -2 -5 -9 10 - h9-9 9 spl - -9 -10 spl spl spl - spl h h8-8 sp 1 sp 1 h/- h/- h/- spl spl spl spl spl spl -2* h h7-7 h/- h/- h/- h/- h/- -9 spl spl spl spl spl h h h6-6 h h h h h/- spl spl spl spl spl 2* h h h5-5 never split - playas total of ten4-4 h h h h h -8 spl spl 3 2 -9* -6* h h3-3 h h h h h -7 -9 spl spl spl spl -6* h h2-2 h h h h h -6 -8 spl spl spl spl h h h

soft 21 8 5 3 1soft 20 2 0 -2 -5 -9 hsoft 19 - -2 -5 -7 dbs dbs - h hsoft 18 h h h h 0 0 0 0 0 h h hsoft 17 h h h h h db db db db db h h h hsoft 16 h h h h h db db db db db h h h hsoft 15 h h h h h db db db db db h h h hsoft 14 h h h h h -8 -10 db db db h h h hsoft 13 h h h h h -5 -7 -10 db db h h h h

hard 21 -hard 20 - hhard 19 - h hhard 18 - h h hhard 17 -6 -9 h h h hhard 16 0 0 -3 -6 -8 - h h h hhard 15 5 5 2 -2 -4 - h h h hhard 14 9 10 7 1 -1 h h h hhard 13 h h h 9 1 -10 - h h h hhard 12 h h h h 8 -9 - h h h h

11 -6 -4 -1 h h db db db db db h h h h10 -3 -I 3 h h db db db db db 8 h h h9 6 h h h h -7 -8 db db db h h h h8 h h h h h 0 -3 -6 -10 db h h h h7 h h h h h 5 0 -2 -6 db h h h h6 h h h h h 7 2 0 -4 db h h h h5 h h h h h 8 2 0 -4 db h h h h4 h h h h h 8 3 0 -4 db h h h h

DOUBLE EXPOSURE 177

Table 78 ContinuedPlayer's Dealer's TotalHand 7 8 9 10 11 12 13 14 15 16 17 18 19 20

double allowed after splitA-A -7 -6 -6 -5 h spl spl spl spl spl 5 6 8 910-10 2 0 -2 -5 -9 10 - h9-9 7 spl - - -10 spl spl spl spl - spl h h8-8 spl spl h/- h/- h/- spl spl spl spl spl spl -2* h h7-7 6 h/- h/- h/- h/- spl spl spl spl spl spl h h h6-6 h h h h h/- spispispIspisP1 2 * h h h5-5 never split - playas total of ten4-4 h h h h h -8 -10 spl spl spl -9* -6* h h3-3 h h h h h spl spl spl spl spl spl -6* h h2-2 3 h h h h -9 -10 spl spl spl spl h h h

KEY

db

dbs

hh/db

h/-spl

number

*

Stand.


Double down; if you cannot double, then stand.

Hit.

Hit or double; never split.


Sp lit.


Split if the count per deck is less than the number;else hit.

Dbl exp halvesWin rate: $31Std dev: $447

Dbl exp hi-optWin rate: $26Std dev: $412


Table 79Halves, Double Exposure, S17


double not allowed after splitA-A spl spl spl -4 -4 -5 -6 -810-10 7 6 6 10 99-9 -2 -4 -3 10 6 3 2 08-8 spl spl spl 7 2 0 -1 -27-7 1 -1 -4 h/- h/- 9 7 56-6 0 -3 -6 h/- h/- 9 7 55-5 never split - play as total of ten4-4 never split - play as total of eight3-3 3 0 -1 h h h 9 82-2 3 0 -1 h h 9 7 6

soft 21soft 20 7 6 6 10 9soft 19 5 3 3 10 8 8soft 18 -2 -5 -7 3 1 7 5 2soft 17 2 -1 -6 h h h 9 7soft 16 4 0 -4 h h h 10 9soft 15 5 1 -1 h h h h 10soft 14 6 2 1 h h h h hsoft 13 6 3 2 h h h h 10hard 21hard 20hard 19hard 18hard 17 -5 -6 -8hard 16 -3 -4 -6 -8hard 15 -1 -2 -4 -6 -9hard 14 -7 -9 -10 0 0 -2 -3 -6hard 13 -5 -7 -7 2 1 0 -1 -4hard 12 -3 -5 -5 6 4 2 0 -111 -10 db db 0 -1 -2 -4 -610 -7 -9 db 5 2 0 -1 -39 0 -1 -3 h 10 7 6 48 8 6 4 h h h h h7 h h h h h h h h6 h h h h h h h h5 h h h h h h h h

DOUBLE EXPOSURE


double allowed after splitA-A spl spl spl -4 -4 -5 -6 -810-10 7 6 6 10 99-9 -3 -5 -5 9 5 2 1 08-8 spl spl spl 6 2 0 -2 -37-7 -2 -4 -8 h/- h/- 7 4 26-6 -3 -5 -8 h/- 10 6 4 15-5 never split - play as total of ten4-4 5 1 0 h h h h h3-3 0 -2 -5 h h 8 6 42-2 0 -2 -4 h 9 7 5 3

KEY

179

db

h

h/-

spl

s17

number

Stand.


Hit.


Split.




Table 80Halves, Double Exposure, HI7


double not allowed after splitA-A spl spl spl -4 -5 -5 -7 -8 -310-10 7 6 5 10 99-9 -2 -4 -6 7 3 1 0 -18-8 spl spl spl h/- 6 3 1 0 spl7-7 1 -1 -4 h/- h/- 9 7 5 h6-6 0 -3 -6 h/- h/- 8 6 4 h5-5 never split - play as total of ten4-4 never split - play as total of eight3-3 3 0 -2 h h 10 8 7 h2-2 2 0 -2 h h 9 7 5 h

soft 21soft 20 7 6 5 10 9soft 19 5 3 1 9 8 7soft 18 -2 -5 dbs h h 8 5 3soft 17 2 -1 -6 h h h 8 6 hsoft 16 3 0 -5 h h h 9 8 hsoft 15 4 1 -2 h h h 10 8 hsoft 14 6 2 0 h h h 10 9 hsoft 13 6 3 1 h h h 10 9 h

hard 21hard 20hard 19hard 18hard 17 -9 -10 3hard 16 -6 -7 -9 6hard 15 -4 -5 -7 -9 9hard 14 -8 -9 -2 -3 -4 -6 -8 hhard 13 -6 -7 -9 0 -1 -2 -4 -6 hhard 12 -3 -5 -8 3 1 0 -2 -4 h

11 -10 db db 0 -1 -2 -4 -6 010 -7 -9 db 5 2 0 -1 -3 29 0 -1 -3 h 10 8 6 4 h8 8 6 4 h h h h h h7 h h h h h h h h h6 h h h h h h h h h5 h h h h h h h h h

DOUBLE EXPOSURE 181


double allowed after splitA-A spl spl spl -4 -5 -5 -7 -8 -310-10 7 6 5 10 99-9 -3 -5 -7 6 3 1 0 -28-8 spl spl spl 10 5 2 0 -1 spl7-7 -2 -4 -8 h/- 10 6 4 2 h6-6 -3 -5 -9 h/- 10 6 4 I h5-5 never split - play as total of ten4-4 4 1 -2 h h h 8 7 h3-3 0 -3 -6 h h 7 5 3 h2-2 0 -2 -5 h 9 6 4 2 h

KEY

db

h

h/-

hI7spl

number

Stand.


Hit.



Split.



The average bet sizes for table 81 are $26.47 for thehigh-low and $26.80 for halves. These are initial betsonly, and do not reflect additional amounts wagered ondoubles, splits, insurance, etc.

Table 81Double Exposure Win Rates

High-LowRaw Dill

Double exposure benchmark 27BJ pays 3:2 91 64Resplits allowed 30 3Double after split 37 10Player BJ beats dealer BJ 33 6Dealer hits soft 17 17 -1 0Double 9, 10, & 11 1 -26Double 10 & 11 -11 -3842 cards not used 32 562 cards not used 25 -2Basic strategy with counting

for bet variation 26 -1no split J-Q 17 -10

HalvesRaw Dill

3198 6735 439 837 621 -10

4 -27-8 -3935 426 -5

31 022 -9

183

CHAPTER 14

FINER POINTS

Expected WinHow much money you can expect to make is a

function of many things. Your expected win is proportional to your bet size and to the amount of your tabletime. The more that you bet the more you will win onlucky streaks and the more you will lose on unluckystreaks. Since you must be financially as well as emotionally prepared to handle an unlucky streak, the morethat you bet the more you must take with you to thecasinos and the more you must have as total capital.

I once got a call from a fellow who wanted to take$500 to Reno for a weekend, and win a couple ofthousand dollars. He asked which book of mine heshould buy to accomplish his goal. I told him sorry, but


to win a couple of thousand in a weekend requires biggerbets than can be justified with a $500 bankroll.

Finding Win Rate Per HourThe win rates in the little rectangles sprinkled

throughout this book are reasonable estimates of whatyou can win in an hour, if you are willing to bet up to $100in the best situations. If you play against the benchmarkrules, with the benchmark betting scheme, and arewilling to make $100 bets on high counts, you can expectto win $16 an hour. If you want to win at a faster rate, youmust:

a) Make bigger bets in those situations whereyou have an edge, or

b) Find a game with better rules, orc) Find a game with fewer decks, ord) Find a game with better penetration, ore) Play more hands per hour.Table 82 summarizes the information in most of the

rectangles throughout this book. It does not containdouble-exposure win rates, and it does not contain winrates for some of the rarer rules. It also contains manywin rates that do not appear in rectangles elsewhere inthis book.

Every number in table 82 was generated by simulation of at least 30 million hands of blackjack. That meansthe standard error on every number in table 82 is lessthan $1. In many cases the sample sizes are more than100 million. Even with samples sizes of 100 million,differences of $1 are too small to be significant.

Benchmark rules for one, two, and six decks are eachbased on samples of more than 600 million hands, andeach has a standard error of $0.20. Thus the two-deckwin rate of $19 is significantly higher than the six-deckwin rate of $16. Both are one deck cut off

FINER POINTS 185

Table 82Effects on Win Rates

One Deck Two Decks Six DecksRaw Dill Raw Dill Raw Difl

Benchmark 48 19 16Dlr hits soft 17 42 -6 13 -6 11 -5Double after split 51 3 23 4 20 4Double 10 & 11 only 38 -10 12 -7 10 -6Double 9, 10, & 11 44 -4 16 -3 12 -4Late surrender 54 6 23 4 21 5Early surrender 77 29 42 23 39 23Lose all to dlr BJ 42 -6 15 -4 11 -5No insurance 41 -7 16 -3 14 -2No resplits 47 -1 18 -1 15 -1Over/under 13 78 30 31 12 32 16Bustout 96 48 42 23 30 14Plr BJ beats dlr BJ 59 1 I 28 9 26 10Dealer's 10 up + ace is

ordinary 21 56 8 25 6 24 86-card win 51 3 22 3 18 2Winning 5-card 21

wins double 50 2 21 2 17 15-card half win 63 15 33 14 30 14Double on any number of cards, including

after splits 61 13 28 9 26 1042 cards not used (21 on

single deck) 59 11 27 8 20 462 cards not used (31 on

single deck) 33 -15 10 -9 13 -3Hi-opt count 40 -8 13 -6 12 -4Halves count 50 2 20 1 17 1Basic strategy with high-low

for bet variation 37 -11 14 -5 12 -4


You can use table 82 to find win rates for combinations of things not considered as a group in this book. Forexample, if you are playing blackjack at a casino thatallows double after split and late surrender and also hasits dealers hit soft seventeen, simply combine the appropriate win rates. There are several ways to do this, andif you have done the arithmetic correctly all methodsshould come up with the same answer: $20 per hour. Oneway is to start with the benchmark $16, add $4 fordouble after splits, add $5 for late surrender, and subtract $5 for dealers hitting soft seventeen.

The average bet sizes for table 82 are $31.48 for onedeck, $27.66 for two decks, and $26.54 for six decks.These are initial bets only, and do not reflect additionalamounts wagered on doubles, splits, insurance, etc.

RiskThe discussion of risk should not be separated from

the discussion of expected win. If you want to doubleyour expected win you have to bet twice as much, andthat means having ups and downs that are twice aslarge.

Those little rectangles sprinkled throughout thisbook have numbers for standard deviation as well aswin rate. Those numbers are all around $400. That $400describes the ups and downs that are typical of an hourof blackjack play with $100 maximum bets. Actually, the$400 number is conservative because the benchmarkdescribes making your maximum bet at counts per deckof +4 or more, and such high counts do not happen often.If you are more aggressive on raising your bets, and arewilling to go to your maximum ($100 in this example) ata count per deck of +2, then the standard deviation of anhour of play is around $500 instead of $400. If you go to$1 00 bets at counts per deck of +1 or more, then the

FINER POINTS 187

standard deviation applicable to an hour of your play isover $600. If you go to $100 bets at all positive counts, thestandard deviation is $750. If you bet $100 on allnonnegative counts, the standard deviation is $800.

These numbers are for playing one hand at a time.If you make the same bet on each of two or more handssimultaneously, your risk will be higher yet.

The easy way to put it all together is to relate all thenumbers to your big bet. Your hourly win rate will be10% to 50% of your big bet. The standard deviationapplicable to an hour of play will be four to eight timesyour big bet. More on risk later in this chapter.

Expected ValueExpected value means the average of all possible

outcomes. To find the expected value, find all of thepossible outcomes (positive and negative) and the probability of each outcome's occurrence. Multiply eachpossible outcome by its probability and sum up theproducts. Your sum is the expected value.

For example, suppose you have a natural against anace and you have seen no other cards in a one-deck game.What is the expected value of insurance in this situation? If the dealer has a lOin the hole, your insurance betwins double. If the dealer has any other card in the hole,you lose your insurance bet. The probability of the holecard's being a 10 is 15/49, and thus the probability ofyour winning double the insurance bet is 15/49.

Possible Outcome-1+2

Sum

Probability Product34/49 - 34/4915/49 30/49

1 - 4/49


The sum of the possible outcomes times probabilities is -4/49. If you insure all of your naturals, you loseeight cents on every dollar of insurance. (The abovecalculation is for single deck, but multiple-deck calculations yield the same loss of eight cents per dollar ofinsurance.) The gambler insures a natural to lock in acertain winner; you should be looking at the -8% expected value.

This is a simple example of expected value. For mostblackjack decisions, calculating the probabilities is morecomplicated. The definition of expected value is thesame - the sum of: possible outcomes times the probabilities of those outcomes.

Thinking in Terms of Expected WinLearn to associate expected value with your bet. If

you have a 2% advantage and a $50 bet, think of that asbeing a $1 expected-win situation. In a single hand youare likely to lose or to win $50; but, over many suchhands, you will average $1 in winnings per hand. If thedealer has a 1% advantage and you are betting $50,think of that as giving the dealer 50 cents. Thinking inthis manner will eliminate any temptation to bet largewhen the dealer has the advantage. Whether you won orlost the previous hand does not alter the value of thishand. If the expected value says that you are giving thedealer 50 cents on this hand, then you are giving away50 cents even if you have just gotten a natural or havewon three hands in a row.

The same logic applies to other casino games. Thedealer's advantage on the pass line at craps is 1.4%; $10on the pass line is giving the casino owner 14 cents.Thinking in this manner will take all the fun out ofgambling and help turn you into an investor.

FINER POINTS 189

Suppose you have a 1.5% advantage on a particularhand of blackjack, and suppose you are betting $50; yourexpected win on that bet is seventy-five cents. If you playtwo hands, in total they are worth a dollar and a half.Your bankroll will go up and down in increments of $50and $100; but, over the long haul, your average win willbe about 75 cents per $50 hand in that situation. Youmust fight one hand at a time, but the object is to win thewar; the war is decided by expected values.

After you have mastered the strategy, a satisfyingway to defme the battle is by use of a target number ofhours played rather than a target number of dollarswon. To play the target number of hours is to win thebattle.

RiskYou can win at this game, but you will not win every

session. An expected-win rate of $16 per hour does notmean that you will win exactly $16 in any given hour.Far from it. You are as likely to lose $399 in an hour asyou are to win $431 in an hour; each is a deviation of onestandard deviation from your expected win. Resultsbetween those two numbers are what you can expect tosee about two-thirds of the time. It will not be rare to seea swing of two standard deviations, e.g. losing $814 in anhour or winning $846 in an hour. That is the risk with nobets over $100, and those big bets coming only on countsper deck of +4 or more.

Suppose you play blackjack with the benchmarkbetting scheme for 40 hours. What is the standarddeviation that describes your ups and downs? Theanswer is easy to find: Simply multiply the one-hourstandard deviation by the square root of the number of


hours you play. For forty hours, multiplying $415 timesthe square root of 40 yields $2625 as the standarddeviation. For the expected win, multiply the number ofhours times the expected win per hour. For 40 hours atan expected win of $16 per hour, the total expected winis $640. An interpretation of these numbers is in 40hours of blackjack with the benchmark rules, two-thirdsof the time your result will be between a loss of $1985and a win of $3265.

Finding Standard DeviationStatisticians use standard deviation and variance

to describe the dispersion of possible outcomes aroundthe expected value. One way to find the variance is tosubtract the expected value from each possible outcome,square these differences, multiply each square, in tum,by its probability of occurring, and sum. The sum iscalled the variance, and the square root of the varianceis called the standard deviation.

Recall the insurance example, in which the insurance bet is lost if the dealer's hole card is other than 10,and double money is won if the hole card is 10. Theexpected value was found to be -8%. The standarddeviation for this example is calculated as follows. (ev isexpected value, sq is square, and prob is probability.)

possible outcome sq prob productoutcome less ev

-1 -.92 .85 34/49 .59+2 2.08 4.33 15/49 1.32

Sum 1 1.91

Each possible outcome is listed. From each possibleoutcome, the expected value is subtracted. Those differ-

FINER POINTS 191

Table 83Area in the Tail of a Normal

Distribution

z = outcome less expected value, then divided by thestandard deviation

shaded shaded shadedZ area Z area Z area.0 .50 1.0 .16 2.0 .023.1 .46 1.1 .14 2.1 .018.2 .42 1.2 .12 2.2 .014.3 .38 1.3 .10 2.3 .011.4 .34 1.4 .08 2.4 .008.5 .31 1.5 .07 2.5 .006.6 .27 1.6 .05 2.6 .005.7 .24 1.7 .04 2.7 .004.8 .21 1.8 .036 2.8 .003.9 .18 1.9 .029 2.9 .002

ences are squared. Then they are multiplied by theirprobabilities. Then those products are summed.

The variance is found to be 1.91. The standarddeviation is the square root of the variance, or 1.38. Overa large number of decisions of this type, say, more than


a couple of hundred, the distribution of possible outcomes approaches the normal.

Normal CurveThe normal distribution is useful. It is symmetric

and has a standard shape. You can tell from the normaldistribution as shown in table 83 how likely a givenoutcome is if you have an expected value and a standarddeviation. The expected value plus or minus one standard deviation contains the actual outcome about 68%of the time. The actual outcome will be worse than onestandard deviation below the expected value 16% of thetime, and the actual outcome will be better than onestandard deviation above the expected value 16% of thetime.

Here is a sample use of the normal curve. What isyour chance of being down by more than $1000 after fourhours of play for the benchmark rules and bettingscheme? Your expected win is $16 per hour, or $64 forfour hours. To be down by $1000 instead of up by $64 isto be $1064 below your expected result. The standarddeviation applicable to your play is $415 for an hour ofplay, and it goes up with the square root of the numberof hours. So the standard deviation for four hours of playis $830. Dividing the amount away from the expectedresult by the standard deviation gives the number ofstandard deviations you are away from the expectedresult: $1064/$830, or 1.3. Table 83 says that the area inthe tail beyond 1.3 standard deviations is .10, meaningthere is a 10% chance that you will be behind by $1000or more after four hours of benchmark play.

Living With RiskWhen deciding how much to bet, pay special atten

tion to the standard deviation because it quantifies the

FINER POINTS 193

risk you will be taking. You must be able to sleep with therisk. You will have streaks where you lose 20, 30, or 40big bets over the course of several days of playingblackjack. If you do not ride easy with such unwholesome and sticky burrs, you will have some grisly dreams.I once lost $5,500 betting $100 per hand on the final dayof a four-day outing, which cut down the trip's win from$12,000 to $6,500. This was back in the 1970s, when$6000 bought a nice car. If you cannot stand the thoughtof losing several thousand dollars, do not bet $100 perhand. Start small. You must build confidence in thesystem and in your ability. Getting a few dollars aheadwill help.

You must get accustomed to thinking of money as ameans of keeping score rather than for what it will buy.Think about buying later. Compartmentalize such thinking away from professional matters. To professionalblackjack players, betting $100 per hand and ending up$6,500 ahead after having been up by $12,000 meansbeing 65 bets up after having been up by 120 bets. If youronly impression upon reading these dollar figures wasthat I was two automobiles ahead after three days andon the final day dropped one automobile to finish oneautomobile ahead, you are not yet ready to bet bigyourself. If you thought of the nwnber of bets instead ofthe purchasing power and you have the required capital,then go ahead and bet big.

Bankroll AdviceFirst, decide how much money you are willing to risk

at blackjack. By that I mean the amount that if you lostit all you would give up the game. That is your total stakefor the purposes of this discussion.

The bet size that is mathematically optimal isdiscussed later in this chapter. Personally, I prefer to bet


more conservatively than can be justified by mathematics. This paragraph presents the approach to bet sizingI use myself. I prefer to relate the biggest bet I am willingto make to my total stake. When I want to play one handat a time, I hold my big bet to 1/150 or less of my totalstake. When I want to play two hands at a time, I holdeach of two simultaneous big bets to 1/200 or less of mytotal stake.

So when I was willing to risk $30,000, I was willingto bet up to $200 on one hand, or $150 on each of twohands.

For a short trip to casino country, a day or a weekend,a sufficient bankroll is 50 times whatever you select asyour big bet. If you want to bet up to two hands of $100,then take $5000 with you (or be able to write checks ofup to $5000.)

For an extended blackjack trip, I recommend takingat least 100 times whatever you select as your big bet.

Doubling your bet size doubles your expected win,but also doubles risk and doubles capital requirements.You can only increase your expected win without increasing capital requirements by playing more hours.Playing two weekends a month will yield twice theexpected win (and twice the expenses) on the same tripcash and the same total stake.

If you want to make a lot of money playing blackjack,you are going to have to play a lot and bet a lot. You mustbe able to lose a lot of money. You do not want to lose, butyou will lose frequently. You will win frequently too, andover the long haul, you are certain to win more than youlose, but the long haul may be a long time in coming. Yousimply must be secure enough financially and emotionally to handle losses that will occasionally amount to the

FINER POINTS 195

trip-cash numbers and may, in a very bad streak, add upto your total stake.

The smaller your big bet in relation to your bankroll,the lower the probability that you will be wiped out.

Your blackjack bankroll behaves much like a portfolio of stocks. A mathematician would describe it asfollowing a random walk with an upward drift. Yourpossible outcomes from investing in blackjack are normally distributed. Investors in the stock market do nothave any assurance that their outcomes are normallydistributed; their possible outcomes appear to comefrom a distribution that has a greater chance of extremely good or extremely poor outcomes.

NadirTo fmd out how far in the hole $50 top bets could put

you in a few days of blackjack, I have run four simulations. This is a rewrite of an article I wrote for the March1990 issue of Blackjack Forum.

The rules used are benchmark plus double aftersplits and no resplits. Other sets of rules would yieldsimilar results.

There was one player at the table, who used thehigh-low with strategy numbers from -1 to +6. Whenever the count went negative, the player switched tables.On counts per deck of zero to Wlder +1, the bet was $5.On counts per deck of +1 to under +2, the bet was $20. Oncounts per deck of +2 or more, the bet was $50. Differentbet schemes would yield different results.

The study was done for both six and eight decks.With six decks, the shuflle came with a deck and a halfleft. With eight decks, the shuffie came with two decksleft.

1000 hands were played, and a record was kept ofthe lowest point the accumulated winnings (or losings)


reached during those hands. Then the bankroll wasbrought back to its initial position, another 1000 handswere played, and another lowest point was registered.This procedure was repeated until 20,000 lowest pointshad been tallied. This was done for both six and eightdecks. Then the whole procedure was repeated, but with4000 hands played before the lowest point is recordedand the bankroll was reset. Thus a total of 200 millionhands of blackjack were played for this study.

Results. The results are shown in table 84. Eachrow shows the probability of being down by a givenamount or more at the lowest point of the session. Forexample, after playing 1000 hands at eight decks, 19.7%of the time your lowest point would have been a loss of$1000 or more.

The data in table 84 are simulation results; eachtrue frequency could be slightly higher or lower. Thestandard errors on the numbers in table 84 range fromalmost zero to 0.35%. You can find the individual standard errors by taking the square root of p(l OO-p)/20000,where p is a number from table 84. For example, thestandard error on the 19.7% is the square root of(19.7)(80.3)/20000, which is 0.3%.

Having an edge means you will be ahead for certainif you play long enough, but it does not guarantee a winin the short run. 4000 hands is a short run.

Comparison of the four columns of table 84 showsthat there is more risk in playing 4000 hands than inplaying 1000. No surprise there. A surprising result isthat risk seems to depend almost entirely on the numberof hands played, and almost not at all on whether theplay was against six or eight decks.

There was slightly more risk playing against sixdecks than playing against eight. That is because there

FINER POINTS 197

Table 84Probabilities of Low Points

Most dollars 1000 hnds 1000 hnds 4000 hnds 4000 hndsever behind 6 decks 8 decks 6 decks 8 decksnever behind 1.0% 1.0% 0.6% 0.5%$1 or more 99.0 99.0 99.4 99.5$100 or more 87.2 86.5 91.9 92.0$200 or more 77.5 76.0 85.6 85.6$300 or more 68.2 66.4 80.0 79.6$400 or more 59.6 57.5 74.5 74.2$500 or more 51.7 49.5 69.0 69.0$600 or more 44.4 42.1 64.2 63.7$700 or more 37.6 35.3 59.5 59.0$800 or more 31.8 29.3 54.9 54.0$900 or more 26.6 24.3 50.4 49.5$1000 or more 21.9 19.7 46.4 45.3$1100 or more 17.9 15.9 42.6 41.6$1200 or more 14.8 12.7 39.0 37.8$1300 or more 12.0 10.2 35.5 34.4$1400 or more 9.6 7.9 32.2 31.2$1500 or more 7.6 6.2 29.3 28.3$1600 or more 6.0 4.8 26.4 25.5$1700 or more 4.7 3.6 23.7 23.0$1800 or more 3.7 2.7 21.6 20.7$1900 or more 2.9 2.0 19.5 18.6$2000 or more 2.3 1.4 17.7 16.6$2100 or more 1.7 1.0 16.0 14.7$2200 or more 1.2 0.8 14.5 13.0$2300 or more 0.9 0.5 13.0 11.7$2400 or more 0.7 0.4 11.7 10.3$2500 or more 0.4 0.3 10.4 9.0$2600 or more 0.3 0.2 9.3 8.0$2700 or more 0.3 0.1 8.0 7.0$2800 or more 0.2 0.1 7.0 6.2$2900 or more 0.1 0.1 6.0 5.5$3000 or more 0.1 0.0 5.2 4.7$3500 or more 0.0 0.0 2.6 2.2$4000 or more 0.0 0.0 1.3 1.0$4500 or more 0.0 0.0 0.6 0.4$5000 or more 0.0 0.0 0.3 0.2$5500 or more 0.0 0.0 0.1 0.1$6000 or more 0.0 0.0 0.0 0.0


were more opportunities to make big bets against sixdecks. Big losses come from being unlucky on big bets.The expected win is greater for six decks, one and a halfcut off than for eight decks, two cut off; that greater profitopportunity greatly outweighs the increased risk.

Likewise, though there is more risk the longer youplay, there is more expected win the more hands youplay.

Remember that table 84 is for bets ranging from $5to $50. Higher bets are accompanied by more risk, butalso of course by more expected win. Table 84 is based onbets of $5, $20, and $50 at specific counts per deck; moreconservative betting would be accompanied by less risk.Table 84 is based on one person at the table, playing onehand at a time. More people at the table probably wouldnot change the risk much, but playing more than onehand at a time definitely would result in more risk.

Trip cash requirements. Table 84 says that for1000 hands, a bankroll of 50 big bets has a risk of ruinof about half a percent. For a play of 4000 hands, astarting bankroll of 90 big bets is required to reduce therisk of ruin to less than half a percent. Those numbersassume no cutting back in bet size in response to losses.

For pessimists. The odds are excellent that sometime in the near future your playing bankroll will be lessthan it is at this instant. For example, table 84 showsthat there is a 99% chance that at some point in the next1000 hands your bankroll will be less than it is right now.

How far is your bankroll likely to fall if your trip issort of average? From table 84 you can find the lowpoints that occur with about a 50-50 chance for the $5 to$50 bet scheme of this study. For 1000 hands, a dip of$500 is about average. For 4000 hands, a dip of $900 is

FINER POINTS 199

about average. Remember that these are worst-pointintermediate results, and not end-of-trip numbers.

Other statistics. The expected wins (simulationresults) for the four columns of table 84 are $226, $167,$904, and $668. The standard deviations are $1028,$960, $2056, and $1920. If you want, you can comparethe nadirs with the final distribution of results asdescribed by the normal curve.

Comments On RiskPeter Giles says:

The late Ken Uston was once quoted in the ReviewJournal as saying, "It's really tough to make a living atblackjack. The fluctuations will really wipe out theaverage guy. If I had to play by myself (instead of on ateam), I probably wouldn't be in it now." You can quoteme as saying the same.

I have chosen to not use an optimal betting system.I would rather start with enough units to have a verylow risk factor and keep the bets the same. I am tiredof having to make up losses by winning more bets thanI lost. I know the reverse is also true, but it still has anegative effect on players on a team. I know one teamthat lost $45,000 in two days, lowered its bets, andground it out for weeks. The players would show upevery day in a bad mood, and it may well have affectedtheir games. Cutting their bet sizes meant they had towin twice as many bets to get back to even in dollars.

I am still trying to determine how many units one issafe with. What is generally recommended in books is,in my opinion, too risky. This is one area in which it ishard to trust mathematics.

Donald Schlesinger says:

I have the utmost respect for both Ken Uston andPeter Giles, but their pessimistic outlook on the individual card counter's plight is, in my opinion, too harsh.


Fluctuations are inherent in the game. Precise andcareful money management protects the average guyfrom the wipeout that Kenny describes. The game canbe beaten and fluctuations can be overcome. Yourreaders should take heart, keep the faith, and practice,practice, practice.

Marvin French says:I am back from Reno after a satisfying trip. I started

by winning, kept on winning, and ended on an all-timehigh. Your advice to stay emotionally detached fromwin/loss results is excellent, but difficult for us humansto follow. I enjoy the game much more when mybankroll goes steadily upward during the trip ratherthan ending high but with huge swings getting there.

Chance of Reaching All-TimeHigh

Unfortunately, most of the time you seem to belosing. According to mathematician Peter Griffin, atmost 1.6% of the time your present hand puts you at anewall-time high. The other 98.4% or more of yourplaying time, your current bankroll is lower than it wasat some time in the past. This can be tough on yournerves. If you compare your present position with yourall-time high, the feeling that you are losing will hauntyou during 98.4% of your playing time. One way to avoidgetting depressed is to forget about your all-time high.Chances of moving up are always greater than chancesof moving down, but downward movements will continue to occur. Your trend is still up. You must hit a newhigh sometime if you keep playing well.

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Avoid Going Broke: UseProportional Betting

By now you have probably won several hundreddollars betting small and are wondering whether youshould increase your bet size. Perhaps a proportionalbetting scheme has occurred to you; that is, you areconsidering betting a fixed proportion of your capital oneach hand.

Some proportions are better than others. The proportion that is best is called the optimal proportion, andwhen multiplied by your total stake is called youroptimal bet size.

In blackjack, the outcome on one round is virtuallyindependent of the outcome of the next round. (TheGwynn and Seri paper presented at the Fourth Gambling Conference found a negative correlation too slightto be of any importance.) This independence is a desirable property in that it simplifies calculation of anoptimal bet size.

If you make a series of optimal bets, you will nevergo broke and you will become wealthy as quickly aspossible. (In theory. In reality, overhead expense andtable minimum leave ruin a serious possibility.) Anyother bet size involves a longer expected time to becomewealthy and may even include a chance of going broke.

Optimal Bet SizeAssuming that you start playing blackjack with a

given amount of capital, make no addition to your totalstake except for blackjack winnings, and make no subtractions from your total stake except for blackjacklosses, your optimal bet size is a proportion of your total


stake equal to your advantage divided by the varianceof possible outcomes.

Your advantage changes from round to round. Atmost casinos the dealer has an advantage over you onthe first round after a shuflle. Therefore, you would liketo bet as small as possible on the first round. If the countstays zero or goes negative, the dealer still has anadvantage over you and if you stay at that table youwould like to bet as small as possible.

If the count goes positive you may have an advantage. For every unit increase in the count per deck, youradvantage increases 0.5% (at standard blackjack; atdouble exposure, the number is 0.7%). To estimate youradvantage, take half of the count per deck and subtractthe dealer's initial edge. Example: A) You are playing ina game where the dealer has 0.5% advantage over youon the first round. Suppose the count per deck is now +5.Your advantage is about 2% (-0.5 + 5xO.5).

Variance of possible outcomes depends on the particular rules you are facing, as is shown in table 85.Being able to double after splitting means higher variance (bigger ups and downs). Having double downrestricted to only ten or eleven means lower variance.

Risk also depends on the number of simultaneoushands you are playing. The measure of how likely twohands are to win 'together or lose together is covariance;it also is shown in table 85. (A covariance of zero meansthe two hands are independent. If you could sit betweentwo blackjack tables and play one hand on each, thecovariance between those two hands would be zero. Acovariance of 1.00 means two hands win and lose in lockstep with each other.)

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Table 85Variance and Covariance for

Blackjack

VarianceCovariance

Benchmark1.280.47

Double Double10 & 11 only After Split

1.20 1.320.43 0.48

The numbers in table 85 for variance and covariance are simulation results generated by BlackjackCount Analyzer.

If you play more than one hand at the same time,your optimal bet size changes because two hands at thesame table are not independent.

Optimal bet size can be found with the aid of table86. To find your optimal bet size, multiply your totalstake times your percent advantage times the appropriate amount from table 86. For example, with $10,000 intotal stake, benchmark rules, and a 1% advantage, youroptimal bet size is $78 on one hand, or $57 each on twohands, or $45 each on three hands. If the count goes upenough so that your advantage is 2%, your optimal betsize will be $156 on one hand, or $114 on each of twohands, etc.

The numbers in table 86 are based on the varianceof one hand and covariance between two simultaneoushands. Letting v be variance, c be covariance, and nbeing number of simultaneous hands, the numbers intable 86 are 1/(v+(n-l)c). This formula is based on the


Table 86Optimal Bet as a Proportion of

Your Advantage

Simultaneous Bench- Double DoubleHands mark 10 & 11 Only After Split

1 0.78 0.83 0.762 0.57 0.61 0.563 0.45 0.49 0.444 0.37 0.40 0.365 0.32 0.34 0.316 0.28 0.30 0.277 0.25 0.26 0.24

trade-off of risk and return popularly known as theKelly criterion.

If you want to remember round numbers for sizingyour bets, perusal of table 86 suggests 80-60-45. That is,for one hand bet 80% of your advantage times your totalstake. For two simultaneous hands, bet 60% of your edgetimes your total stake on each. For three simultaneoushands, bet 45% of your edge times your total stake oneach.

Note that playing two hands instead of one shouldmean betting only 50% more money. If consideration ofrisk and return suggests betting say $200 on one hand,then if you want to play two hands you ought to bet $150on each for $300 total. If you want to go to three hands,you ought to bet around $112 on each for $336 total. Sothe total money you can justify betting on one rounddoes not increase much if you want to play three or moresimultaneous hands. The reason is the covariance - the

FINER POINTS 205

results of all your simultaneous hands are tied to thedealer's hand. Suppose you play two simultaneoushands. A natural for the dealer means you are likely tolose both of your hands, and if the dealer starts with asmall card up and busts you likely will win both of yourhands.

Your advantage varies from round to round, andvarying your bet size as your advantage varies is theoretically desirable. As a practical matter, however, varying your bet size with your advantage will not make youa millionaire; it will get you barred from play. To avoidthat cruel fate, you must blend in with the gamblers.

How much you should bet is determined by morethan just what is mathematically optimal. If you are notemotionally prepared to accept the risk associated withoptimal bets, then bet less. If you bet less than theoptimal amount, you will never go broke, but you will notget rich as fast as you might. Know yourself - checkyour emotional readiness to lose before you set up abetting plan.

You must be cold and calculating. Steely blue eyeswill do. You must have no emotional attachment to thatmoney you throw on the table. It must mean little to you.If you lose it, so what; it is only money. If you cannotafford to lose it, you cannot afford to bet it. In no way isshort-run profit guaranteed. You cannot plan on winning the next hand. Lady Luck has a heart of stone. Youwill be way ahead if you play long enough, but only if.The actual play is a hand at a time and you must beemotionally able to handle the losses that will occurregularly. Bet an amount you can live without. You mustbe able to withstand a one-day loss of twenty timeswhatever is your big bet, because such a loss will occuroccasionally.


Some bet sizes are more convenient than others. Forexample, suppose your optimal bet size is $9.16. Now$9.16 is an impossible bet size for blackjack. Even $9 isan awkward bet size. Go ahead and bet $10. Roundingoff does not hurt. The important thing in optimal bettingis to be sure to reduce your bet when your capitalshrinks.

OverbettingPerhaps your attitude toward risk is that you are

willing to wager more than the optimal amount on eachhand. For example, maybe your capital is $1,000 but youare willing to use $25 as your big bet.

If you bet more than the optimal bet size, you areoverbetting. Overbetting will cause you to win morewhen you are lucky and to lose more when you areunlucky. To avoid going broke, you would have to cutyour bet size after a series of losses. The disadvantage ofoverbetting is that the extra win when you are lucky isoutweighed by the accelerated bet cuts dictated by thebigger losses when you are unlucky. You can overbet fora while and get away with it, but the person who bets theoptimal amount consistently will make more moneyover the long haul than the person who overbets consistently.

Overbetting can be financially fatal. It can be provedmathematically that an overbettor who consistentlybets twice the optimal amount will break even over thelong haul. The overbettor who bets more than twice theoptimal amount will go down instead of up in the longrun. If you continually overbet you will go broke eventhough you have an advantage. Do not overbet.

FINER POINTS 207

If you do not understand why overbetting is bad,deal 100 hands to yourself and pretend bet, say, onefourth of your capital on each round. You must recalculate your capital after each round. Your capital goesdown fast, even if you win more hands than you lose.

Stopping RulesA cousin problem of overbetting is what to do when

you are losing and running out of pocket cash. Alwayskeep enough money to split a pair or to double down. Forexample, if you are down to your last $200 and the packis very profitable, the most you should bet is $100, or $50for each of two hands. Save the other $100 in case youmust double down or split a pair. Try to carry enoughcash so that you do not get into this position. Never makea bet that you would be unwilling or unable to match ifyou should have to split a pair or double down. You onlyhave an advantage over the dealer if you are willing andable to follow the recommendations of the system.

Although a time comes during a losing streak whenyou should stop, namely, when you run out of pocketcash, no rule for stopping governs a winning streak. Theprobability of your winning the next hand from anhonest dealer is the same whether you have been losingor winning. Quitting offers no advantage when you area certain number of dollars ahead. True, it is disappointing to make only $6,500 on a trip when you were $12,000ahead at one time, and you may tend to blame yourselffor being greedy; you tell yourself that you should havequit with the $12,000. Cheer up - you did the rightthing in continuing to play. No optimal stopping rulesexist. You never know whether you are right now at ahigh point or whether you will continue to win in the


hours ahead. If you are playing correctly, the chances ofyour winning another thousand dollars always exceedthe chances of your losing another thousand dollars.Never become apprehensive, and never become satisfied.

If you think you are being cheated, then of coursestop betting. Your chance of being cheated in a legalcasino in Nevada or Atlantic City is close to zero.

The Optimal Number ofSimultaneous Hands

Optimal bet sizes, including zero when you do nothave an edge, are based on mathematics and computersimulations. These are the amounts you would like tobet if you could bet any amounts you want. But of courseyou cannot bet anything you want - your bet size issubject to constraints. There are minimum and maximum bets. Only certain bet sizes are accepted. If casinoemployees are scrutinizing you to evaluate your ability,you might impose additional constraints on yourself. Onnegative counts: If you cannot bet zero, then bet theminimum you can get away with. If you play a spot, thatspot gets cards. If you leave that spot blank, the dealerdoes not give it any cards. Playing one more hand thususes more cards, and may mean fewer rounds before theshuffle. For example, if enough cards remain for threemore head-up rounds of one hand each, you will get onlytwo rounds if you play two hands each round. Whichmakes more money, three single hands or two doublehands? The answer depends on how much you arebetting per hand. It also assumes that the dealer will nothave to shuffle until at least one subsequent hand. (Ifthis is the last round before the shuffle no matter how

FINER POINTS 209

many hands you play, then optimally if you have an edgeyou should spread your money over as many hands aspossible.)

Alone and High LimitsIf you are alone at a table, if you have an advantage,

if your optimal bet size for one hand is less than the tablemaximum, and if another round will be dealt after thisone, then it does not matter much whether you bet onehand or two, but do not go to three hands. To go to twohands of optimal bets is to bet about 150% as much perround as to play one hand per round. And to add a secondhand is to use about 150% as many cards. You will get 2/3as many rounds before the dealer shuffles, but you willbe betting 150% as much per round, so two hands at atime has the same profit potential as one hand perround.

Adding a third hand and making optimal-sized betswould add 12% to your total bets but would use 33%more cards, and thus result in lower overall profitpotential.

If you are the only customer at a table and the dealerhas the advantage, your optimal action if you stay is toplay one hand at the minimum. To play two minimumbets at a time would be to bet twice as much on each ofthe two-thirds as many rounds before the shuffle, for33% more unprofitable action.

Therefore, if you are the only customer at the table,it makes sense to play one hand at a time whether or notyou have the advantage. This advice holds if youroptimal one-hand bet size is less than the table maximum.


Alone and Low LimitIf you are the only customer at the table and your

optimal single-hand bet is more than the table maximum, then your optimal total bet on two simultaneoushands will be more than 150% of the table limit. In thiscase, you will be better off with two hands per round. Youwill get two-thirds as many rounds before the shuffle,but you will be betting more than 150% as much perround.

One Other Player and High LimitIf there are other players at the table with you, your

adding an extra hand adds a smaller proportion to thetotal cards used on one round. You would bet 50% morewith two optimally-sized bets instead of one, but withone other player you would use only 33% more cards perround. This means you are better off playing two handsif there is another customer at your table and you havean edge.

Do not go to three hands of your own; your thirdhand would use 25% more cards but your total bets (withoptimal bet sizes) would increase by only 12% per round.

More Other PlayersWith two or more other hands played besides your

own, you are better off with two hands yourself whenyou have an edge. However, increasing to three hands ofoptimal-sized bets will not win you more money nomatter how many other players are in action at yourtable.

Another Argument for Fewer HandsYour optimal bet size changes as your advantage

changes, and you would like to estimate your advantageas closely as possible. Playing fewer hands means estimating your advantage more often between shuffles,

FINER POINTS 211

hence more precisely. For example, suppose you areplaying two hands at once, and suppose your first handis 10-10. Your second hand can be thought of as comingfrom a pack that is poorer by two 10s than the pack youused to estimate your advantage to select your bet size.Had you known in advance that those two lOs would notbe available to your second hand and ,to the dealer, youwould have bet less on your second hand. The existenceof this precision factor suggests being hesitant aboutplaying another spot when a few cards could cause bigchanges in the count per deck. For example, if one deckis used, you might stick with one spot if you are alone orthere is one other player, and switch to two spots if thereare two or more other players.

SummaryGenerally, if you are alone with the dealer you are

better off playing one hand, and if there are otherplayers you are better off playing two hands when youhave an edge.

All Decisions AreApproximations

Every counting system involves approximations.Whatever the high-low system tells you is your advantage is only an estimate of your true advantage. Yourtrue advantage might be slightly more or less than whatany count indicates. When the high-low system recommends a certain playing-strategy option, that option isprobably (but not always) the best way to play thatparticular hand.

Some more complicated counting systems are capable of making slightly better approximations than the


high-low system, but only if they are used absolutelycorrectly. Many users of more complicated countingsystems switch back to the high-low after discoveringthat increased complexity leads to increased errors incasino play, and these errors more than offset any gainfrom using the more complex count.

Even a computer makes approximations. All counting systems, even computers, assume that the next cardto be dealt is equally likely to be any of the unseen cards.A computer can make better decisions than you canbecause it can keep track of how many cards of each rankremain unseen. A computer is capable of manipulatingthese data to make the best decisions possible given thatthe unseen cards can be in any order, but such calculations are time-consuming. For much faster but onlyslightly poorer decisions, a computer that keeps track ofhow many of each rank remain unseen will still useapproximations to arrive at its recommendations.

Recently some brilliant people have taught computers to keep track of the cards in the order used, calledshuffle tracking. These computers can do better thansimply assume the next card is equally likely to be anyone of those yet unseen.

The simulation results reported in this book are notthe best a computer can do. What I asked my computersto do is play blackjack using the counting systems andthe strategy numbers presented in this book. Thus thewin rates presented in this book are attainable by ahuman. Though a computer could make decisions basedon the exact mix of cards remaining rather than on acounting system and strategy numbers, that has notbeen done for any of the simulation results reported inthis book.

FINER POINTS

Which Counting System

213

A major decision a card counter faces is whichcounting system to use. Here is a letter from a readerthat expresses the quandary:

I hope you can help me. I am writing to you for someadvice. I have been playing blackjack for about a yearnow and I have been counting cards for most of thattime. I love the game and plan to play semiprofessionally, but here is my problem. What is the best countmethod to use?

When starting to play last year, I read Humble andCooper's book, The World's Greatest Blackjack Book,and I used the Hi-Opt I count. Next I read the lateLawrence Revere's Playing Blackjack as a Businessand started using his Hi-Lo because it was not an aceneutral count. I played blackjack in Las Vegas usingboth counts on different visits and won money usingboth. A few months ago, I read Bryce Carlson's Black-jack for Blood, and of course he says his basic andadvanced Omega counts are best. And then, to makematters more confusing, I read your book, BlackjackSecrets, and you use the Hi-Lo count, but you vary thestrategy according to the count and the dealer's hittingor standing on soft seventeen.

I know that there are different levels of counts, butwhich is the best?

I consider myself of at least average intelligence andI do not mind putting in the hours of study that it takesto become a professional blackjack player. I just do notwant to waste any more time on a blackjack systemthat is not going to help me be the best player possible.

So, there is my dilemma. If you could help me, itwould be much appreciated.

My answer to this reader is: It does not matter muchwhich counting system you use, as long as you are using


something that compares high cards to low cards. Thehigh-low is the best of the simple systems. If you want touse a more powerful system, the halves count winsfaster than the high-low, but is also more difficult.

Much more important is what kind of a game youcan get, specifically how much you can bet on highcounts. I suggest that instead of trying to learn a morecomplicated counting system or memorizing a widerrange of numbers than -1 to +6, you would be better offthinking about what kind of an act would allow you toget away with bigger bets when your count finds anedge.

Analyze what kind of image you project (or fixyourself up to project the image of your choosing) andthen select an act that is consistent with the image youproject. You can have multiple acts, each with its ownlook (clothing, hair, personality, et cetera).

Getting away with card counting in a casino is a bigand important topic, and it has its own book: BlackjackSecrets.

How should aces be counted? According to PeterGriffm, the importance of each card for betting purposesis close to the importance of each card in detection ofappropriate deviations from basic strategy, except foraces. The high-low and halves systems presented in thisbook count aces in the high-card group. This is better forbetting purposes, partly because naturals pay three totwo.

Some other systems, hi-opt for example, count ace aszero in the primary count. For most playing-strategydecisions, more accuracy is achieved if aces are countedas zero. However, for hi-opt to approach the high-low inwin rate, a side count of aces must be incorporated forbetting purposes. Keeping a side count means keeping

FINER POINTS 215

two running counts, which of course is much moredifficult than keeping just one running count.

If you use only one count (Le. no side count of aces),should aces be counted in the high group or counted aszero?

My answer is to include aces in the high group, withthe lOs. The reason is that estimating your edge for betsizing is more important than knowing when to varyfrom basic strategy. The simulation results reported inthis book provide ample evidence. My simulations ofsimple counting systems (one count only, and no sidecount) show that the high-low outperforms the hi-optwhether naturals pay even money or 3:2. The high-lowyields a higher win rate than the hi-opt for any numberof decks. The sample sizes are large enough that thedifferences are statistically significant.

Halves does better than the high-low, $1 per hourbetter for the benchmark rules, but is also more difficultto use. Whether you think the improvement in win ratejustifies the increase in difficulty is a personal matter.Ever since I developed it in the early 1970s, I have usedthe halves count with no side counts. For most of myplaying career I used strategy numbers from -1 to +6.

How Many Numbers To LearnThe text of this book presents strategy numbers

from -10 to +10. The appendix presents strategy numbers from -20 to +20. I personally use strategy numbersfrom -1 to +6. What should you use?

Part of the reason I memorize such a limited rangeof strategy numbers is during the days I supportedmyself playing blackjack, I traveled the world to find thebest games, and I played against quite a variety of rules.


I learned the strategy numbers appropriate to whateverrules I was facing at the time. Often on a plane flight Iwould learn the strategy numbers appropriate to thecasinos of my destination. For example, sometimes Icombined Korea with Macao; I would learn the appropriate Korea numbers on the plane to Seoul, and thenreplace them in my brain with the Macao numbers onthe flight from Seoul to Hong Kong.

Another reason for learning a small range of numbers is you are fooling yourself if you think you reallyneed a large range of numbers. Simulations show littleif any increased win rate for memorizing strategy numbers less than -1 and greater than +6. As for negativestrategy numbers: You ought not to be playing againstnegative counts, and if you do have to play against themyou ought not to be betting much, so why worry aboutdeviating from basic strategy? As for high positivestrategy numbers: They occur rarely in an honest game.If you do happen to get a high-low count per deck of say+lOin a shoe game, you ought to be asking yourselfwhether you are getting an honest game rather thanrejoicing in a rare opportunity to use high strategynumbers. My own experience is that the only times Ihave gotten a count per deck of +10 or more on each ofthe first two shoes I played, I later found out I was beingcheated.

I am including strategy numbers from -10 to +10 inthis book because people want to buy a wide range ofstrategy numbers. I can generate them, and I canpublish them for you, but that does not mean I recommend that you use all of them.

Wm rates for the range of numbers -1 to +6 are only$1 per hour less than the win rates for the benchmarkrange of -10 to +10.

FINER POINTS 217

The high-low system involves a simple countingrule and a relatively small amount of data to be memorized; yet it gets the money. You can estimate youradvantage more accurately, but only by using a morecomplicated counting system. You can learn a moreaccurate playing strategy, but only at the cost of memorizing more strategy numbers.

Side Count of AcesThe 1981 and earlier editions of this book contained

strategy numbers for high-low adjusted for aces andhalves adjusted for aces. Using those numbers requiredkeeping a side count of aces. That material is notincluded in this edition due to its lack of importance inincreasing your win rate.

Two-Card CombinationsThe tables in the text of this book base playing

strategy on total points in your hand and not on whichparticular cards make up your hand. Appendixes A andB contains information on strategy for particular twocard combinations.

The additional dollars per hour you can win byknowing two-card strategy numbers is essentially zero,particularly with multiple decks.

The most important two-card combination is 7-7against 10 for a single deck. With the high-low system,for instance, you should stand with 7-7 against 10 withone deck if the count per deck exceeds -1, whereas the


first part of table Al (page 255) says that you shouldalways hit fourteen against 10.

You might examine the two-card combinations toget an understanding of how sensitive certain decisionrules are to the removal of key cards. For example,thirteen against 2 is sensitive to 7s and 8s. The greaterproportion of 7s and 8 that are left in the deck, the morelikely you should be to hit thirteen against 2. The greaterproportion of 7s and 8s that you have already seen, themore likely you should be to stand on thirteen against 2.

Resolving Toss-upsHere is a tip for decisions when whether or not to

take another card is a toss-up. If the count is negative,take the card; if the count is positive, do not take the card.The reason for this advice is: The more cards that remainto start the next round, the more likely it is that therewill be a next round.

Quick and AccurateYou will never learn to play perfectly like a com

puter. The computer will spot correct deviations frombasic strategy that you would never catch. For example,a computer can spot situations in which splitting 3-3against lOis the correct decision; all that is necessary isa big excess of 8s and 9s in the remainder of the pack. Thesystems in this book say do not split 3-3 against 10,because a count of 8s and 9s is useful for few otherdecisions. You can never playas well as a computer, justalmost as well.

FINER POINTS 219

To win in casinos you must make decisions quicklyand accurately. It is not advisable to use a countingstrategy in a casino if you cannot play quickly andaccurately with it. No strategy modification will yieldbig gains over the high-low system. If you want to playmany hours in a day, you cannot use a strategy modification that you find tiring. Keeping two counts is moretiring that keeping a single count, and keeping threecounts is considerably more tiring.

You can modify your system from day to day or hourto hour, since all of the decision tables are similar. Theuse of one set of decision tables with a different countingsystem or a different number of decks results in fewincorrect decisions.

I like to use a simple counting system so that I canplay long hours, talk with other people at the table, keepan eye on the pit boss, and not get exhausted. When I fIrststarted playing blackjack, I used Thorp's 10-count, whichinvolves keeping two running counts. I made moneywith it, but it was a lot of work. Then I switched to thehigh-low, and not only did I make money faster, but theeffort of counting seemed to be reduced by 90%. Now Iuse the halves count, which is only a bit more work thanthe high-low and (for me) worth the extra effort.

Thorp's Complete Point CountYou may have compared my high-low strategy num

bers with the complete point count system of Ed Thorpin the second edition of Beat the Dealer and wonderedwhy the differences exist. For a valid comparison, cutThorp's numbers in half because his high-low index isapproximately equal to count per two decks. Thorp'srules assume one deck, with the dealer standing on soft


seventeen, and no restrictions on doubling down. Thus,you might compare Thorp's numbers with table A3 (page263). The numbers are not identical, and the mainreason is the way Thorp treated 7s, 8s, and 9s. Supposethat Thorp wanted to create a deck that had a count of+5; he would do so by removing five small cards from afull deck. The remaining 47 cards would thus be rich inaces and lOs, but would also be rich in 7s, 8s, and 9s. Thisprocedure distorted some frequently-used decision rules.A person who is using Thorp's rules would tend to hittwelve against 2 too often and would stand on sixteenagainst 7 too often.

With Other People at the TableA reader says:

The count a twenty-one player uses represents thecomposition of cards not seen, which many times, whennot playing head-on, is different from the compositionof the cards remaining to be dealt. Since count systems,as I understand, are developed on the basis of head-onplay, will not the playing efficiency of any system besignificantly affected when playing with other people?If so, should a player attempt to guess the unseen cardsin the other players' hands and then adjust the countfor that one hand? This is an area no one seems todiscuss; therefore I hope you will answer my questions.

The playing efficiency of your count system is notaffected by playing with other people at the table.Playing strategy does not depend on how many playersthere are at the table. The main effect of having moreplayers is slower play; fewer hands per hour means alower hourly win rate. Yes, you should try to guess the

FINER POINTS 221

unseen cards in other players' hands and use thatinformation in the play of your hand.

Penetration is important. Look for games with goodpenetration. And develop an act that allows you to betmore money when you have an edge than when thecasino has an edge.


CHAPTER 15SHUFFLES

One way to get an edge is to follow individual cardsor groups of cards through the shuffle. Most casinoshuffles are so complicated that attempting this is awaste of time.

Shuffles are standardized within a casino; whatever shuffle you witness on one table will likely berepeated on other tables that use the same number ofdecks.

Occasionally you find a casino in which the dealersuse an incomplete shuffie. That is, some shuftles involvebreaking the pack into half-deck-size groups and shuffling those groups two at a time, but otherwise notintermixing the groups. If you count cards, you might beable to have a count for each of several different groupsof cards, follow those groups through the shuffie, and cutan excess of aces and lOs to the top.

SHUFFLES 223

New Decks Shuffled, But NotTogether

Something to watch for if you start a game at atable with unshuffled cards: See if the dealer mixes allthe decks together. Sometimes each deck will be shuftledindividually, and then the decks will be slapped together(but not shuffled together) and given to you for a cut.

Bill Gaem popped into Pop's Oasis in Jean, Nevada. The dealer was standing at the table. In front ofhim were spread four decks. They obviously were usedcards, but they had been sorted and arranged to resemble new decks. Then a customer came. The dealershuffled each deck individually, but the individual deckswere not shuffled with each other. Thus the shoe contained four single decks, stacked one on top of the other.

Another method of shuffling new cards, observedat a five-deck game in the Riviera in Las Vegas, isintermixing two decks, then intermixing two more, andlastly shuffling one deck by itself. If you see a dealershuffle like that, you can watch where the cut card isplaced to see whether you are playing against one of thedouble decks or a single deck.

To a basic-strategy player, it makes little differencewhether a shoe contains single decks stacked one on topof the other or intermixed decks except that the casino'sedge is about half a percent smaller with the singledecks. But to a card counter the difference is major. Arunning count of +4 in a single-deck game calls for a bigbet, whereas a running count of +4 early in a four-deckgame is still in the ho-hum range.


Single-Deck Shuffle StudyThis is a study of the most common shuffle used at

single-deck blackjack in Las Vegas casinos. The sampleconsists of ten independent shuffles, each of whichstarts from the same arrangement of cards.

Sampling MethodologyDave Douglas, who was a blackjack dealer in Las

Vegas casinos for five years, did the shuflling. I askedhim to do typical casino shuffles - neither overlycareful nor overly sloppy. The shuflling was done on astandard blackjack table at the home of Peter Giles.Casino cards were used. The resultant shuflles are thusrepresentative of what you routinely encounter in singledeck games in casinos.

The cards were initially arranged in the order ofnew decks: by suit, with the ace of hearts on top. Douglasperformed ten shuffles, each of which began with thesame ordering of the cards, and each of which includedthe same steps.

The shuffle used is rift1e-riffle-strip-riflle. A variety of shuftles are actually used in casinos, and this oneis common. It is also about the minimum you find incasinos. Many dealers riffle more than three times orstrip more than once. Only rarely will you encounter adealer who riffles fewer than three times or does notstrip.

A riffle is breaking the deck into two approximately equal parts and interleaving one into the other.Dealers also call this a "shuffle" but I am using "riffle" toavoid confusion; I am using the word "shuffle" to meanthe sum total of the actions to reorder the cards. In thisstudy, the "shuffle" is riffle, riftle, strip, riflle.

SHUFFLES 225

A strip is breaking the deck into six to ten parts andinverting the order of the parts. The dealer holds thedeck with one hand and pulls groups of cards off the topwith the other hand.

Most shuflles also include turning half of the deckaround during one of the riffles, but turning the cardsend for end does not change their order.

DataThe shuftle data are presented two ways. Table 87

lists the cards by suit and point value. Examples: DA isdiamond ace, CT is club 10, H7 is heart 7. Table 88 hasexactly the same data except that each card has beenreplaced by its ranking in the deck before the shuffle.

The first column of both tables is labeled "0." Itcontains the order of the cards before each of the tenshuffles. Each of the columns labeled 1 to lOis thecolumn 0 deck after a riffle-riffle-strip-riffle sequence.

Analysis of ResultsYou might think that the same dealer doing the

same shuffle on identical decks would rearrange thecards approximately the same way every time. Comparing columns 1-10 with each other shows that this iscertainly not the case. Each shuffle results in a differentfinal arrangement of cards.

You might be wondering if any particular cardwould be more likely to end up in a predictable spot. Atest of this is to see how often two shuffles put the samecard in the same location. The probability of a card in oneshuffle being matched by the same card in the samelocation in another shuffle is 1/52. There are 52 locationstotal, and 45 pairs of comparisons possible per location.Multiplying the above three numbers together yields45, which is the expected total number of times a card in


Table 87Raw Data From Ten Shuffles

0 1 2 3 4 5 6 7 8 9 10HA OA C4 H7 07 OK OT C2 CK C4 OJH2 CT H7 S6 S7 S6 C9 OA S6 OK H9H3 H7 HJ C6 C6 sa H3 08 S2 02 SAH4 S6 S7 C7 C2 DQ sa C3 H4 H8 saH5 C5 C9 OT OK C5 OA H8 HJ H9 H4H6 08 C5 H4 S6 CK H9 C4 DQ OA D4H7 CK 03 C3 C7 C2 C2 H9 SK HQ C7H8 HT OQ 55 DQ S5 CT C5 OJ CJ S7H9 55 CT 05 C3 C8 S7 SK HQ 07 OTHT C6 OJ 09 C8 SQ CJ 07 C7 SK S9HJ 07 02 D4 H2 S4 09 H5 SQ S3 03HQ S4 sa S4 05 C9 CQ 06 H2 CQ H3HK OK H6 H2 C4 OJ DB H6 SA S2 C6CA H3 OA C8 H5 D4 CK H7 C8 HK HTC2 SK S9 HQ OJ 03 S6 02 HK DB SJC3 C7 HQ HK D4 CA C3 C6 OT SQ C3C4 OQ C3 OK S5 02 H4 OJ CA CA C4C5 D6 SA ST H6 S3 SA C9 09 SA HJC6 HJ H4 C9 OT OA OJ CA SJ SJ STC7 sa CJ S9 CJ H9 H2 CT C9 H5 C5C8 C4 SJ CT 03 SK 02 S7 C2 sa 02C9 HQ CQ DQ HK H5 S9 D4 H6 CT DQCT S7 H5 CA SA S2 C7 OT HA H6 CACJ so CK 03 sa D6 CA CJ 55 HJ C2CQ H2 ST sa CA HQ HA 09 H9 S7 H7CK OT C8 OJ H4 SA H8 HA 03 S6 HKOK SJ C2 H8 CQ 05 HT CQ OK 06 coDQ ST HT CQ C5 OT SQ S9 02 H7 SQOJ HK D4 H9 CK HK C8 H4 S4 55 H2OT CA OK HT 06 HT HJ S3 HT 05 H809 02 S2 SK OA 09 C5 HK S3 OQ S4DB SA SK 02 S4 HA SJ S2 OA D4 OK07 C2 08 HJ SK 08 D4 OK DB H4 S306 CJ H9 C5 H3 H2 HQ SA H5 C5 OA05 H8 SQ CK S3 87 DK sa H3 C6 05D4 H9 S3 S2 H9 H3 S2 03 C4 OJ HQ03 S9 CA SQ D9 H8 DQ C7 C3 HT 8202 CQ 07 SJ SQ H4 C6 OQ ST 03 SKOA C3 HA DB 08 C6 ST H2 05 OT HASK H4 C7 CJ S2 HJ 03 SJ CT HA 55SQ D9 H2 HA HT C7 HK 05 C5 C7 D6SJ HA HK SA SJ 07 H7 C8 S9 H2 CKST S3 H3 08 HJ H6 SK HQ H8 S4 CJS9 OJ DB C4 ST SJ S5 H3 CJ H3 09sa C8 05 S3 S9 C3 07 S6 D4 C8 C8S7 05 S6 OA C9 CT C4 55 sa 09 C9S6 H5 S5 H5 HQ 5T 06 CK C6 C9 H555 S2 H8 07 CT C4 H5 S4 CQ C2 S6S4 D4 C6 H6 02 CJ S4 ST S7 ST H6S3 H6 OT C2 HA CQ H6 HT 08 CK CTS2 C9 09 S7 H7 59 S3 so H7 C3 08SA 03 S4 H3 H8 H7 05 HJ 07 S9 07

SHUFFLES 227

Table 88Card Positions of Ten Shuffles

0 1 2 3 4 5 6 7 8 9 101 39 17 7 33 27 30 15 26 17 292 23 7 47 46 47 22 39 47 27 93 7 11 19 19 45 3 32 51 38 524 47 46 20 15 28 45 16 4 8 455 18 22 30 27 18 39 8 11 9 46 32 18 4 47 26 9 17 28 39 367 26 37 16 20 15 15 9 40 12 208 10 28 48 28 48 23 18 29 24 469 48 23 35 16 21 46 40 12 33 3010 19 29 31 21 41 24 33 20 40 4411 33 38 36 2 49 31 5 41 50 3712 49 45 49 35 22 25 34 2 25 313 27 6 2 17 29 32 6 52 51 1914 3 39 21 5 36 26 7 21 13 1015 40 44 12 29 37 47 38 13 32 4216 20 12 13 36 14 16 19 30 41 1617 28 16 27 48 38 4 29 14 14 1718 34 52 43 6 50 52 22 31 52 1119 11 4 22 30 39 29 14 42 42 4320 45 24 44 24 9 2 23 22 5 1821 17 42 23 37 40 38 46 15 45 3822 12 25 28 13 5 44 36 6 23 2823 46 5 14 52 51 20 30 1 6 1424 41 26 37 45 34 14 24 48 11 1525 2 43 45 14 12 1 31 9 46 726 30 21 29 4 52 8 1 37 47 1327 42 15 8 25 35 10 25 27 34 2528 43 10 25 18 30 41 44 38 7 4129 13 36 9 26 13 21 4 49 48 230 14 27 10 34 10 11 50 10 35 831 38 51 40 39 31 18 13 50 28 4932 52 40 38 49 1 42 51 39 36 2733 15 32 11 40 32 36 27 34 4 5034 24 9 18 3 2 12 52 5 18 3935 8 41 26 50 46 27 45 3 19 3536 9 50 51 9 3 51 37 17 29 1237 44 14 41 31 8 28 20 16 10 5138 25 33 42 41 4 19 28 43 37 4039 16 1 34 32 19 43 2 35 30 140 4 20 24 51 11 37 42 23 1 4841 31 2 1 10 20 13 35 18 20 3442 1 13 52 42 33 7 21 44 2 2643 50 3 32 11 6 40 12 8 49 2444 29 34 17 43 42 48 3 24 3 3145 21 35 50 44 16 33 47 36 21 2146 35 47 39 22 23 17 48 45 31 2247 5 48 5 12 43 34 26 19 22 548 51 8 33 23 17 5 49 25 15 4749 36 19 6 38 24 49 43 46 43 650 6 30 15 1 25 6 10 32 26 2351 22 31 46 7 44 50 41 7 16 3252 37 49 3 8 7 35 11 33 44 33


one shuffled deck is in the same location in anothershuffled deck if the cards are completely random afterthe shuftle. An example is C4 is the top card in shuftles2 and 9. The actual total of such pairs in the ten shuftlesis 48. This can be treated as a Poisson distribution,meaning the variance is approximately equal to theexpected number of pair~, or 45; hence the standarderror on the actual number of pairs is the square root of45, or 6.7. The actual number of pairs is close to theexpected number. Therefore knowledge of the bottom (ortop) card before the shuffle is useless information afterthe shuffle (assuming the absence of eyesight keenenough to follow the card through the shuffle).

Does the above paragraph mean the shuffle provides a random reordering of the cards? Not necessarily.A random reordering means that every card has anequally likely chance of appearing at any position afterthe shuftle, which seems to be the case, but it also meansthat every arrangement of cards is equally likely.

Here is a test of whether some arrangements ofcards are more likely than other arrangements. Look atthe pairs of cards that are initially adjacent, and analyzetheir relative positions after the shuffle. For example,before the shuffie CJ (club jack) follows CT (club 10).After shuffle #1, CT is the second card and CJ is thethirty-fourth card; the number of cards after CT andbefore CJ is 31. Treat the shuftled decks as if the top cardfollows the last card; this will happen when the deck iscut. Example: H7 initially follows H6. After shuffle #1,H6 is the fiftieth card and H7 is the third card. After thecut there will be four cards after H6 and before H7.

Table 89 lists the final distribution of cards thatwere adjacent before the shuflle. "Gap" is the number ofcards between initially adjacent cards. A gap of zero

SHUFFLES 229

Table 89Post-Shuffle Gaps Between

Initially Adjacent CardsGap Freq Gap Freq Gap Freq

o 23 17 5 34 111 57 18 3 35 112 40 19 4 36 93 43 20 9 37 104 35 21 3 38 85 23 22 3 39 76 18 23 6 40 87 6 24 1 41 138 9 25 5 42 109 6 26 8 43 6

10 3 27 7 44 311 1 28 6 45 612 5 29 7 46 413 6 30 13 47 014 6 31 8 48 215 1 32 10 49 016 2 33 19 50 1

means the cards are still adjacent and in the same order.A gap of 50 means the cards are adjacent after theshuffle, but the order is reversed; the only time thishappened in the sample is in shuffle #8: C4 ended updirectly ahead of C3.

If the order of the cards were truly random afterthe shuffles, the theoretical distribution of the gapbetween initially adjacent cards would be uniform. Thatis, a random shuffle would find the 51 0 data points


spread over the 51 possible gaps at the rate of about 10per gap.

The sample shows an excess of small gaps betweeninitially adjacent cards. The chi-square value is 646.4with 50 degrees of freedom, which corresponds to a zstatistic of 26. Therefore you can say beyond doubt thatthe distribution of cards after the shuffle is not random.Knowledge of the exact order of the cards before theshuffle has value in predicting the very next card to bedealt because small gaps between initially adjacentcards occur more often than could be due to chance.

No large clumps of cards were preserved throughany of the shuffles. There was not a single instance ofthree cards staying next to each other. In a completelyrandom shuffle the probability that three cards remaintogether (or are separated and then reunited) is once perfifty decks, or 0.2 per ten decks; this study got zero, whichis not significantly different from expectation.

In a completely random shuffle the probability ofthree initially-adjacent cards finishing up in that orderwith no more than one card separating them is 0.8 perten decks; this study got one, which is not significantlydifferent from expectation.

Practical ConsiderationsI receive many letters discussing shuffling, and

most of the comments can be put into two categories.Some people are concerned that in some diabolic way thestandard casino shuffle works to their disadvantage.There is no evidence in this study to support such anotion.

Other people think that the standard casino shuffleis so poor that it can be exploited by players who haveknowledge about the order of the cards before theshuffle. In this study, 47% of the time cards that were

SHUFFLES 231

adjacent before the shuffle were in the same order andseparated by six or fewer cards after the shufile. Perhapsthis could be exploited by a player to gain an advantage.For example, if you memorize the card atop an ace beforethe shuffle, when that card appears there is a 47%chance its ace will follow within six cards (assuming asingle deck).

Speed of BlackjackSingle-Deck Blackjack

This is an empirical study of how fast single-deckblackjack is dealt in casinos at Reno, Nevada. A stopwatch was used to measure the actual time taken toshuffle the cards and to deal to various numbers ofplayers.

DataI timed a total of 42 shuffies and 167 rounds of play

for a variety of dealers at a variety of casinos. The resultsare shown in table 90.

Shuffle TimeThe time recorded for the shuffle is the total

elapsed time from the instant the dealer broke the deckto the instant the first card was dealt on the next round.Shuffle time thus includes time for cutting the deck andburning a card. Shuffle time also includes time foranything else that interrupted the dealer during theshuffle; the most frequent such interruption was formaking change. One dealer blew her nose during theshuffle, so her nose-blowing time is included. Withoutany interruptions, the mean shuffle time would be justunder 20 seconds. With interruptions included, themean time for the 42 shuflles I observed is 23 seconds.


Table 90Average Times for Shuffling

and Dealing Single-DeckBlackjack In RenoNumber of Sample Size Mean Time

Players (Rounds) (Seconds)

shuffle 42 231 46 172 21 233 23 324 19 335 29 456 21 507 8 59

Dealing TimeDealing time was from the instant the first card

was dealt to the instant the first card was dealt on thefollowing round or to the instant the dealer broke thedeck for a shuffie. There were sometimes short delays,primarily for a player to buy chips; the clock was alwaysrunning dwing these delays. The players' first two cardswere dealt face down. The players could double downonly on ten or eleven. All samples were taken at casinosthat offered insurance. If a dealer ran out of cards beforefinishing a round, that sample was aborted.

Dealing time depends on the speed of the dealerand the speed of the players. Individual hands alsodiffer; e.g. if the dealer has a natural, the hand is overquickly. This study was done back when dealers checkedhole cards under lOs.

SHUFFLES 233

Mean dealing time increased approximately linearly with the number of players. The average round ofsingle-deck blackjack in Reno took about (7n + 10)seconds, where n is the number of player hands.

It appears that the average dealer's hand took 10seconds, or longer than the average player's hand, whichwas 7 seconds, but this interpretation is incorrect. Youcannot separate the time for the dealer's hand from thetime for the first player's hand. The mean combined timefor the dealer and the first player was 17 seconds. Eachadditional player took 7 seconds on average to play ahand. However, you cannot say that the first playeraveraged 7 seconds to play a hand. The reason is secondand later players were studying their hands while thedealer was waiting for the fITst player to make a decision.The first player did not have this extra time.

Estimates of Rounds per HourUsing the empirical estimates for Reno single

deck games of 23 seconds per shuflle, 17 seconds for thedealer and the first player, and 7 seconds for eachadditional player, you can estimate the number of roundsper hour for various numbers of players and variousnumbers of rounds between shuffles. These are onlyaverages of course; the actual number of rounds perhour you will get in a casino will vary somewhat fromtable 91. In particular, if you find a fast dealer and playfast yourself, you can get considerably more rounds perhour in a head-on game.

Discussion of Rounds Per HourA player of average speed with a dealer of average

speed in a head-on game with seven rounds betweenshufiles gets about 177 rounds per hour. When anotherplayer joins the game there will probably be about five


Table 91Average Rounds Per Hour for

Single-Deck Blackjack in RenoPlayer Hands Rounds Between Shuffles

1 234 5 6 7 8

1 90 126 146 158 167 173 177 1812 77 101 114 121 126 1293 67 85 93 984 59 73 79 825 53 64 686 48 577 44 51

rounds between shuffles, and the speed will be cut toabout 126 rounds per hour. When a third player joins thegame, there will be at most four rounds between shuflles;the players will be getting about 98 rounds per hour.When the table fills up, rounds per hour will be cut inhalf again.

Besides slowing play down dramatically, increasing the number of players at a table has another harmfuleffect: It means fewer rounds between shuflles, and thatmeans playing a larger percentage of the hands withfreshly-shuffled cards. The 51 hands per hour at a fulltable are split - half directly after shuffles and halfwhere you get to see some cards before making a bet. Atthe head-on game, the 177 hands per hour include about27 right after shuftles and 150 where you get to see somecards before making a bet. If you disregard the first handafter each shuffle, one average hour at a head-on gameyields as many hands as six hours at a full table. If youplay quickly, the ratio might be higher than six to one.

SHUFFLES 235

The implication of course is that you are muchbetter off with a dealer to yourself than at a full table.

Six-Deck BlackjackThis is an empirical study of how fast six-deck

blackjack is dealt in casinos at Atlantic City, NewJersey. A stopwatch was used to measure the actualtime taken to shuffle the cards and to deal to variousnumbers of players.

DataI timed a total of 13 shuflles and 261 rounds of play

for a variety of dealers at a variety of casinos. The resultsare shown in table 92.

Table 92Average Times for Shuffling

and Dealing Six-DeckBlackjack in Atlantic City

Number of Sample Size Mean TimePlayers (Rounds) (Seconds)

shuffle 13 100

1 59 122 53 203 31 274 28 355 38 386 21 427 31 48


Shuffle TimeThe time recorded for the shuffle was the total

elapsed time from the instant the dealer reached for theused cards to the instant the first card was dealt on thenext round. Shuffie time thus includes time for cuttingthe pack and burning a card. Shuflle time also includestime for anything else that interrupted the dealer during the shuflle; the most frequent such interruption wasfor making change. Shuffles of new cards were notincluded; they are much slower; each changing of thecards took about ten minutes. Different casinos useddifferent shuffles, but they all took about the sameamount of time. The mean time for the 13 shuffles Itimed is 100 seconds.

Dealing TimeDealing time is from the instant the frrst card was

dealt to the instant the first card was dealt on thefollowing round or to the instant the dealer reached forthe used cards to shuffle them. There were sometimesshort delays, primarily for a player to buy chips; the clockwas always running during these delays. The players'cards were dealt face up; the players never touched thecards. Dealing time seemed to depend primarily on thespeed of the dealer. Some dealers were very fast; somewere capable of 500 hands an hour for head-on play.

Mean dealing time increased approximately linearly with the number of players. The average round ofblackjack in Atlantic City took about (7n + 5) seconds,where n is the number of player hands. This is about 5seconds per round faster than single-deck blackjack inReno; the difference seemed to be due to the fact that theAtlantic City game was dealt face up.

SHUFFLES 237

You cannot separate the time for the dealer's handfrom the time for the first player's hand. The meancombined time for the dealer and the first player was 12seconds. The actual number might be slightly less than7 seconds, but my sample size is too small to be moreprecise.

Rounds Per HourUsing the empirical estimates for Atlantic city six

deck games of 100 seconds per shufile, 12 seconds for thedealer and the first player, and 7 seconds or eachadditional player, you can estimate the number of roundsper hour for various number of players. Table 93 assumes the dealer cuts off about 100 cards; dealers whocut off fewer cards will deal more hands per hour. Theseare only averages of course; the actual number of roundsper hour you will get in a casino will differ somewhatfrom table 93.

Table 93Average Rounds Per Hour forSix-Deck Blackjack in Atlantic

CityPlayer Hands Rounds Per Hour

1 2482 1583 1164 915 766 647 56


Eight-Deck BlackjackI timed 21 shuffies of eight decks at various Atlan

tic City casinos. The average time was 2:22 for reshuffiesof used cards. Shuffles of new cards take about fiveminutes.

Atlantic City BlackjackA reader adds:

What prompts me to write is your articles on speedof play in Reno and Atlantic City casinos. I am developing my own rules of thumb for evaluating win ratesper hour at blackjack games. I have noticed that eachplayer takes visibly longer to play a hand at face-downgames compared to face up, mainly because in face-upplay the dealer anticipates the player's actions. So Iwas surprised that both of your articles said sevenseconds for each additional hand; I would have expected the face-up Atlantic City game to be faster thanthe face-down Reno game.

I ran off least squares linear curve fits on your data,weighting for sample size. For Reno I got (6.8n + 10.0),which closely agrees with your (7n + 10). However, forAtlantic City, the result is (6.0n + 7.6), which is quitedifferent from your (7n + 5).

You are correct in that the actual play of the handsis faster at face-up games than at face-down games. Thisfactor tends to make Atlantic City faster than Reno.

There is another factor, however, that works in theopposite direction. The other factor is the time it takesthe dealers to do other things, such as make change.These other things take longer in Atlantic City than inReno. You can bet cash in Nevada, but in Atlantic Cityyou must convert your cash into chips - even fordoubling down and buying insurance.

SHUFFLES 239

The sample sizes in table 92 are too small for muchprecision, but I suspect the underlying relationship isnot linear. Nonlinearity is caused by new customersbuying chips to join in the game. This effect is muchstronger in Atlantic City than in Reno. Games with twoto five players seem to be more vulnerable to beingslowed down by new customers buying chips. Fulltables, of course, cannot have new customers buying inunless a player leaves. Perhaps full tables really averagecloser to 60 rounds per hour instead of the 56 listed intable 93.

Search For StreakinessRandom Shuffle

Several blackjack authors state that the distribution of outcomes of blackjack games is more "streaky"than one would expect from strictly random outcomes.They say that wins come in bunches that can be predicted by an observant player. They say that somedealers are "cold," meaning easier to beat than a dealerwho is "hot." Charles Einstein in his Basic BlackjackBetting advocates "rhythm betting" to take advantage ofstreakiness. Eddie Olsen developed and Jerry Pattersonsold TARGET, a system that in part tries to select tableswhere the cards are biased in favor of the player andwhere the dealer is busting more frequently than usual.These are not the only authors who claim you can profitby predicting streakiness.

Our brains try to see patterns in things that arerandom. Common sense seems to support the notionthat there are runs of good and bad luck. This is one areawhere our brains can play tricks on us. What seems to becommon sense sometimes is nonsense.


None of the authors who say you can make moneyon streakiness has provided evidence to support theexistence of streakiness. Therefore, I have done thisstudy to measure streakiness and the difference between hot and cold dealers to estimate how much moneyyou can make by betting more when you are winningand looking for cold dealers, and how much you can saveby avoiding hot dealers.

StudyThis study uses benchmark plus double after split

and no resplits, eight decks shuftled, six dealt out, andone player using basic strategy. (There is nothing specialabout this set of rules; it just happened to be common inAtlantic City at the time the study was done.) Thecomputer keeps track of whether the player wins, loses,or pushes. It also keeps track of the dealer's final hand.It hits the dealer's hand until it gets to seventeen or moreeven if the player busts or gets a natural, so as to seewhether the dealer is hot. Results are kept in sets ofthrees. The sample size is 20 million.

Results - Win, Lose, or PushTable 94 shows the relative frequencies with which

wins, losses, and pushes occur as a function of how theplayer has fared on the previous two hands. For example, the first row is what happens on the next handafter you have won two hands in a row.

The overall averages are .432 chance of winning,.478 chance of losing, and .089 chance of pushing, whichdo not quite sum to 1.000 due to rounding. Though yourchance of winning is smaller than your chance of losing,your average win when you win is greater than youraverage loss when you lose. You win three to two on yournaturals, and you win your doubled bets more frequently than you lose them.

SHUFFLES 241

Table 94Outcomes Versus Results of

Previous Two Handsprevious outcome of this handtwo hands win lose push

winJwin .432 .478 .089winJlose .432 .478 .089winJPush .432 .478 .089loseJwin .433 .478 .089lose,lose .432 .479 .089lose,push .433 .478 .089push,win .433 .478 .089pushJlose .432 .478 .089pushJPush .432 .479 .089

The chi-square statistic for table 94 is 6 with 16degrees of freedom. This means there is no relationshipwhatsoever between what happens on this hand andwhether you have won, lost, or pushed on the previoustwo hands. Trying to bet more when you are winning orless when you are losing is an exercise in futility; it willnot win any money. Streaks occur, but they cannot bepredicted.

Results - Dealer's handTable 96 shows the relative frequencies with which

dealers get particular totals as a function of what theyhave gotten on the previous two hands. For example, thefirst row shows what happens on the hand after thedealer has gotten two consecutive naturals. 21 means alltwenty-ones except naturals. The overall averages for


dealer totals are shown in table 95. (The total is slightlydifferent from 1.000 due to rounding.)

The chi-square statistic for table 96 is 473 with 288degrees of freedom. This translates to a z-statistic of 7,which means the dealer's outcome is not independent ofwhat the dealer has done on the previous two hands.

Careful analysis of table 96 discloses the nature ofthe relationship. The asterisks denote deviations of 3.5standard errors or more. Without exception, the significant deviations are lower frequencies for the dealergetting a natural after having gotten a natural on one orboth of the previous two hands. Thus this test is sensitiveto the effects of removal of an ace and a 10 out of 416cards! The other deviations from average follow noparticular pattern, and are typical in size of the randomfluctuations you expect to see in a sample of this size.

Though the huge sample size makes this testextremely sensitive, there is no support for the notionthat good hands beget good hands and bad hands beget

Table 95Average Probabilities for

Dealer's Hand, 817

natural .04821 .07320 .18019 .13918 .13417 .145bust .282

SHUFFLES 243

Table 96Dealer's Hand Versus TwoPreceding Dealer Hands

previous hands bust 17 18 19 20 21 naturalnatural,natural .286 .146 .136 .135 .180 .073 .045*natural,21 .283 .145 .139 .134 .179 .075 .045*natural,20 .283 .147 .139 .134 .178 .073 .046*natural,19 .283 .146 .139 .134 .179 .073 .047natural,18 .281 .146 .140 .134 .180 .074 .045*natural,17 .282 .147 .139 .134 .179 .072 .046natural,bust .284 .145 .139 .133 .178 .074 .046*21 ,natural .283 .145 .139 .134 .181 .072 .04621,21 .283 .144 .138 .134 .180 .072 .04821,20 .282 .145 .140 .133 .180 .073 .04721,19 .283 .145 .139 .134 .179 .073 .04821,18 .282 .145 .139 .133 .180 .073 .04821,17 .281 .144 .140 .134 .181 .073 .04821 ,bust .280 .145 .140 .133 .180 .073 .04820,natural .282 .145 .139 .135 .179 .073 .04720,21 .281 .145 .139 .133 .178 .074 .04820,20 .281 .146 .140 .134 .178 .073 .04720,19 .282 .145 .139 .133 .181 .073 .04720,18 .282 .146 .139 .133 .180 .073 .04820,17 .282 .146 .140 .133 .179 .073 .04720,bust .282 .146 .140 .134 .179 .073 .04719,natural .283 .145 .141 .133 .178 .073 .04619,21 .282 .145 .140 .133 .180 .073 .04819,20 .281 .145 .140 .133 .180 .073 .04719,19 .281 .145 .140 .133 .181 .072 .04719,18 .282 .145 .140 .134 .179 .072 .04819,17 .282 .145 .139 .134 .180 .073 .04719,bust .281 .146 .140 .133 .179 .072 .04818,natural .283 .146 .139 .134 .179 .073 .04618,21 .282 .145 .138 .133 .181 .073 .04818,20 .282 .145 .140 .134 .179 .073 .04718,19 .282 .146 .138 .133 .181 .073 .04718,18 .283 .145 .139 .133 .180 .072 .04818,17 .282 .145 .139 .133 .181 .072 .04818,bust .281 .145 .139 .133 .181 .073 .04817,natural .282 .146 .140 .135 .180 .073 .046*17,21 .283 .145 .138 .133 .180 .073 .04817,20 .283 .145 .139 .134 .180 .073 .04717,19 .283 .144 .139 .133 .181 .073 .04817,18 .282 .145 .139 .133 .180 .072 .04817,17 .282 .145 .139 .133 .180 .072 .04817,bust .281 .145 .140 .134 .180 .072 .048bustnatural .281 .146 .140 .134 .180 .073 .047bust21 .282 .145 .139 .134 .180 .073 .047bu5t20 .282 .146 .139 .134 .179 .073 .048bust19 .281 .145 .139 .134 .180 .073 .048bust18 .282 .145 .139 .134 .180 .072 .048bust17 .281 .145 .139 .133 .181 .073 .047bustbust .281 .145 .140 .134 .180 .073 .048


bad hands. Thus there is no such thing as an honestdealer who can be relied upon to stay hot - or to staycold.

Blackjack Without ShufflingThe above section' shows that there is no such

thing as streakiness if the cards are shufiled thoroughly.This section examines the opposite extreme - whathappens if the cards are not shuffled at all.

For cards new from the factory, shuflling is necessary. If a customer at the table is sophisticated enoughto keep track of the location of certain cards or groups ofcards, shuffling of used cards is necessary. But asidefrom that, does shuffling accomplish anything, or is itmerely superstition and show?

Argument For ShufflingThorough shuflling is necessary to get an arrange

ment of cards that behaves in a random fashion.

Argument Against ShufflingThe cards are placed in the discard pile in an order

different from that in which they are dealt. This guarantees that even if the cards are not shuffled, there is noway that the same cards could be dealt to the sameplayers shoe after shoe. It is true that this reordering ofcards is not random, and that some card combinationswill occur with frequencies different from those if thecards were randomly shuffled. This might result inplayers winning slightly faster or slower than if thecards were shuffled, but it should not cause winningrounds to be followed by winning rounds or losingrounds to be followed by losing rounds.

Reordering By DealingDifferent dealing procedures result in different

reorderings of cards. The reordering procedure used in

SHUFFLES 245

this study is called "lay and pay." An alternative procedure is called "pick and pay." Naturals (unless the dealerhas a chance of a tying natural) and busted players'hands go to the discard pile first. In lay and pay, thedealer's hand is next on the discard pile. Then comeunbusted player's hands, with first base on the bottom.If pick and pay were used, the dealer's hand would go ontop.

In the discard pile, the order of the cards in eachplayer's hand is first card received on top and last cardreceived on the bottom.

The order of discards of the dealer's hand is holecard on the bottom, upcard next, and hit cards on top inthe order received, with the last card received on top.

For example, if there is one player against thedealer and neither takes a hit, cards ordered 1-2-3-4before the deal are 1-3-2-4 on the discard pile. For twosuch rounds in succession, cards ordered 1 through 8before the deal are 5-7-6-8-1-3-2-4 on the discard pile.

For one round on which the dealer and player eachtake two hits, cards ordered 1 through 8 before the dealare ordered 1-3-5-6-8-7-2-4 on the discard pile.

Pair splits are similar to multiple players in thatthe hand closer to first base is finished first, and ends upon top of the other hand in the discards. If there is onlyone player, that player splits a pair, and each split handrequires a third card, cards ordered 1 through 8 beforethe deal are 3-7-8-1-5-6-2-4 on the discard pile. Threeplayers, none of whom take hits, would cause cards 1through 8 to be 3-7-2-6-1-5-4-8 on the discard pile.

PurposeThe purpose of this study was to find out how a

player would fare shoe after shoe if the cards wereshuffled only when taken new from their boxes.


MethodologySix decks were shufiled, and one card was burned.

There was one player betting $100 and playing one handof basic strategy against Atlantic City rules (dealerstands on soft seventeen, double after splits, and noresplits). The player's cards and the dealer's cards wereplaced in the discard pile in the same order as lay andpay in a casino. This procedure continued round afterround until four and a half decks had been used, atwhich time the round was finished, the unused cardswere placed on top of the discards, and the pack was cut.The cut was random and no closer than one deck toeither end of the pack.

For each shoe the study kept track of the amountthe player won or lost. Then it compared performanceson consecutive shoes.

ResultsThe results for more than a million shoes (over 44

million rounds) are shown in table 97. The first columnis the win in the previous shoe. The second column is thenumber of shoes with a given win in the previous shoe.The next column is the number of hands in those shoes.After that comes the most important column, averagewin per hand. The last column is the standard errorapplicable to each average win.

Table 97 shows that shuffling is necessary. Therearrangement of cards for the situation tested, namelyone player, lay and pay, and six decks, results in theplayer being 0.70% to 0.75% better off than if the cardswere shuffled. The average payback for one player usingbasic strategy is 100.25%, and the standard error on thisnumber is 0.02%. A player advantage due to not shuffling may sound farfetched, but John Gwynn and ArnoldSnyder came up with the same result in a later study

SHUFFLES 247

Table 97One Player, $100 Bets, Six

Unshuffled Decks, Lay and Paywin, prev shoe shoes hands avg win se

-$1000 or less 97,298 4,147,719 $0.14 $0.06-$950 or -$900 25,579 1,090,278 0.60 0.11-$850 or -$800 30,380 1,294,846 0.32 0.10-$750 or -$700 34,765 1,482,673 0.35 0.09-$650or-$600 39,169 1,669,770 0.46 0.09-$550 or -$500 43,416 1,850,967 0.43 0.08-$450 or -$400 47,343 2,018,584 0.26 0.08-$350 or -$300 50,427 2,150,228 0.28 0.08-$250 or -$200 52,938 2,257,191 0.24 0.08-$150or-$100 54,461 2,321,709 0.25 0.07-$50 or zero 55,377 2,361,017 0.18 0.07$50or$100 55,068 2,347,264 0.16 0.07$150 or $200 54,826 2,337,371 0.37 0.07$250 or $300 52,354 2,231,678 0.21 0.08$350 or $400 49,316 2,102,559 0.29 0.08$450 or $500 45,228 1,927,914 0.18 0.08$550 or $600 41,782 1,781,407 0.29 0.09$650 or $700 37,387 1,593,599 0.18 0.09$750 or $800 32,832 1,399,614 0.28 0.10$850 or $900 28,524 1,216,007 0.30 0.10$950 or more 115,784 4,936,597 0.14 0.05total 1,044,254 44,518,992 $0.25 $0.02

KEY

se Standard error.


("Ruffied by the Shufl1e," Blackjack Forum, March 1990,pages 5-12).

Table 97 indicates that winning or losing in theprevious shoe does not foretell winning or losing in thisshoe, even if the cards are not shuffled and the numberof players does not change. Differences between winrates of the various lines of table 97 are so small thatthey probably are random. There is no such thing asstreakiness if the cards are never shuffled.

Streakiness does not exist at the two extremes,when cards are well-shuffled and when they are notshuffled. Since streakiness does not exist at both extremes, it does not exist between the extremes. That is,streakiness does not exist, regardless of the amount ofshuffling.

Card Combinations Without ShufflingThis section explains how not shuffling except

when the cards are new can give the player an edge.

MethodologyThis study consists of 500,000 consecutive shoes.

The details were the same as in the previous study.At the conclusion of each shoe, the order of the

cards in the discard pile was examined. The frequencieswith which each pair of consecutive cards occurred weretallied.

Then the study was repeated, and the frequencieswith which each pair of alternate cards occurred weretallied.

ResultsTable 98 contains the expected frequencies for any

two cards, including subsequent and alternate, for thorough shuftles. This table is based on mathematics, notsimulation.

SHUFFLES 249

Table 98Any Two Cards from Well-Shuffled Six-Deck Shoes

first relative frequency ofanother cardcard ace 2 3 4 5 6 7 8 9 10

ace .074 .077 .077 .077 .077 .077 .077 .077 .077 .3092 .077 .074 .077 .077 .077 .077 .077 .077 .077 .3093 .077 .077 .074 .077 .077 .077 .077 .077 .077 .3094 .077 .077 .077 .074 .077 .077 .077 .077 .077 .3095 .077 .077 .077 .077 .074 .077 .077 .077 .077 .3096 .077 .077 .077 .077 .077 .074 .077 .077 .077 .3097 .077 .077 .077 .077 .077 .077 .074 .077 .077 .3098 .077 .077 .077 .077 .077 .077 .077 .074 .077 .3099 .077 .077 .077 .077 .077 .077 .077 .077 .074 .30910 .077 .077 .077 .077 .077 .077 .077 .077 .077 .305

Table 99 contains the actual relative frequenciesfor subsequent cards for the sample of 500,000 unshuftled shoes. The simulations were played out for oneplayer head-on against the dealer. Table 100 containsthe actual relative frequencies for alternate cards for thesame sample of 500,000 unshuffled shoes.

There are many differences between tables 98, 99,and 100. All of these differences are statistically significant due to the large sample sizes. The differences arisebecause the play of the hands is not random. Forexample, aces as the first two cards to the player arealways split, which causes consecutive aces to be lesscommon with unshuflled cards than with shuflled cards,.058 to .074. Since split aces receive one and only onemore card, aces with a card between them occur moreoften with unshuftled than shutlled cards, .082 to .074.


Table 99Consecutive Cards From

Unshuffled Six-Deck Shoesfirst relative frequency ofsubsequent cardcard ace 2 3 4 5 6 7 8 9 10

ace .058 .078 .078 .077 .076 .077 .079 .081 .081 .3152 .082 .073 .084 .082 .081 .080 .076 .075 .073 .2943 .082 .084 .072 .082 .080 .079 .076 .075 .074 .2964 .081 .084 .083 .076 .080 .079 .077 .076 .074 .2905 .081 .083 .082 .081 .077 .079 .077 .076 .074 .2916 .081 .081 .081 .080 .080 .071 .078 .077 .076 .2967 .078 .075 .075 .075 .075 .076 .069 .079 .079 .3198 .079 .075 .075 .076 .076 .077 .078 .062 .080 .3239 .079 .074 .074 .075 .075 .076 .078 .079 .070 .32110 .075 .073 .074 .074 .075 .076 .078 .080 .080 .314

In general, the difference between well-shuffledcards and unshuffled cards is that if the cards are notshuffled, high cards tend to follow high cards and lowcards tend to follow low cards.

Shuffle Every Fifty ShoesPeter Griffin asked what would happen with a

shuffle every 50 shoes. So I ran another million shoeswith a shuffle after every 50, and the result is a paybackof 100.23% (with a standard error of 0.02%). This 100.23%is not significantly different from the 100.25% of theprevious study.

Other Numbers of PlayersI also ran simulations of 300,000 shoes each with

various numbers of players and a shuffle every 50 shoes.With two players, the result was a payback of 99.90%with a standard error of 0.03%. With three players the

SHUFFLES 251

Table 100Alternate Cards From

Unshuffled Six-Deck Shoesfirst relative frequency ofcard after subsequent cardcard ace 2 3 4 5 6 7 8 9 10

ace .082 .073 .073 .074 .074 .074 .076 .077 .077 .3212 .077 .075 .080 .081 .081 .079 .076 .075 .074 .3013 .077 .081 .076 .080 .080 .078 .076 .075 .074 .3024 .077 .081 .080 .077 .079 .078 .076 .076 .075 .3015 .077 .080 .080 .079 .076 .078 .076 .076 .076 .3026 .m.~.~.~ m .~ .m .m m .~

7 .m.m.m.~.m.m ~ .m .m .~

8 .078 .076 .076 .076 .076 .077 .077 .076 .078 .3119 .078 .075 .075 .076 .076 .076 .077 .077 .078 .31210 .075 .076 .076 .076 .076 .076 .078 .079 .079 .310

result was a payback of 99.73% with a standard error of0.05%. With four players, the result was a payback of99.72% with a standard error of 0.07%. With five players, the payback was 99.67% with a standard error of0.08%. Thus the payback for blackjack without shuftlingvaries with the number of players at the table; moreplayers means a lower player edge.


APPENDIXES A & B

The appendixes are not recommended reading. Youdo not need them to play blackjack to win. They are forthose readers who want a wider range of numbers thanis presented in the earlier chapters. Each consists of fourtables, identified by letter and number. The letter designates the counting system.

Appendix A - high-lowAppendix B - halvesThe number differentiates the number of decks and

the dealer action on soft seventeen.1 - one deck, dealer hits soft seventeen2 - four decks, dealer hits soft seventeen3 - one deck, dealer stands on soft seventeen4 - four decks, dealer stands on soft seventeenEach table is spread over three pages. The first page

is a table covering hitting or standing, doubling down,and pair splitting. The second page covers hitting orstanding and doubling down by two-card combinations.The third page covers surrender by two-card combinations. Then comes a blank spacer page.

ApPENDIX A 253

To use the numbers in the table, first drop thefraction from your count per deck. Examples: a count perdeck of 5.2 rounds to 5, 2.8 rounds to 2, -2.3 rounds to -2,and -1.9 rounds to -1.

A number in a table means different things, depending on where you are in the table. It means stand, doubledown, split, or surrender if your count per deck is thenumber or more. If your count per deck is lower than anumber in the table, then hit, do not double down, do notsplit, or do not surrender. If a number is followed by anasterisk, the usual meaning is reversed.

The upper limit on early surrender is only if youcannot both surrender and buy insurance. If you can doboth on the same hand, then ignore the numbers thatare followed by an asterisk in the surrender tables. Forexample, look at table A4 for early surrender of 8-8against ace. The table says -14/12*. If the count per deckis less than -14, do not surrender, and play your handaccording to the advice elsewhere in the table. If thecount is between -14 and +12, surrender. If the count perdeck is above +12: Surrender if you can both surrenderand buy insurance, but if doing one means you cannot dothe other then buy insurance and do not surrender.

These tables are my own independent work. Anyresemblance between them and the work of JulianBraun or anyone else is fortunate. My method of calculation is to start with one or four decks, and remove theknown cards (the dealer's upcard and your two cards, ifknown). I then add cards in proportion to the valuesassigned to them by the counting system (for negativevalues that means subtracting cards) to achieve a desired count per deck. Then I assume that I have aninfinite number of those resulting packs. I find theexpected value of each possible play for each hand.


*

**

db

hhI7

h/-

When the best play changes as I change the count perdeck, I have found a strategy number for that hand.

These tables differ from earlier editions of thisbook. The change was increasing all the negativenumbers by 1 because in the 1981 and earlier editionsof Professional Blackjack, strategy numbers werepresented using a different assumption as to how toround count per deck.

Appendix AThe set of tables in Appendix A contains the decision

rules for the high-low count (2,3, 4, 5, 6 = +1; 7, 8, 9 = 0;10, ace = -1). The four tables in this part of the appendixcover both dealer actions on soft seventeen for one andfour decks.


KEY TO APPENDIXES A AND B

Stand.

Double down.

Hit.



hlsp Either hit or split, but never stand.

spl Split.

Sp/- Either split or stand but never hit.

number Stand (or double or split) at a count per deck equal toor greater than the number; hit (or do not split) at acount per deck less than the number.


With 4-4, splitting is better than doubling down onlyif doubling is allowed after splitting.

ApPENDIX A 255

Table AlHigh-Low, One Deck, HI7

D ~ r'. U rdPlayer's eae s peaHand 2 3 4 5 6 7 8 9 10 ace

soft 18 -16 -18 -15 -16 -14 -16 h h 17

hard 17 -19 -19 -5"0 hard 16 -9 -10 -12 -13 -13 11 10 5 1 3j hard 15 -5 -8 -8 -9 -11 12 12 8 5 5en

hard 14 -3 -4 -8 -7 -9 16 h h h 9~

:E hard 13 0 -1 -3 -4 -7 h h h h 15hard 12 3 2 0 -1 -2 h h h h h

11 -11 -12 -13 -14 -15 -8 -5 -3 -3 -110 -8 -9 -10 -11 -13 -8 -3 0 4 39 1 0 -2 -4 -5 3 8

eu 8 14 9 5 2 2 14:c 7 17 11 9 8:s0 6 15 11 12"0

5 15 12 13

ace-9 10 9 7 5 4 15ace-8 9 4 2 1 0 16ace-7 0 -1 -10 -10 -15 18

0 ace-6 0 -4 -8 -14 -15:c:s ace-5 15 4 -3 -7 -150 ace-4 18 4 -3 -6 -10"0

ace-3 12 5 -1 -4 -6ace-2 11 5 1 -1 -3

ace-ace -13 -14 -15 -15 -16 -12 -10 -10 -10 -810-10 11 9 7 5 5 13 20

Q) 9-9 0 -1 -2 -4 -5 6 -14 -12 0:0= 8-8 spl spl spl spl spl spl spl 11* 6* -20 7-7 -14 -16 spl spl spl spl"0

0 6-6 0 -1 -4 -6 -7 0*~ 5-5.t: 4-4** 18 11 7 80.. 3-3 8 4 -1 -4 -5 13*en

2-2 9 5 0 -4 -7 spl

ace-ace -13 -14 -15 -15 -16 -12 -10 -10 -10 -8Q) 10-10 11 9 7 5 5 13 20:0 9-9 -2 -3 -4 -5 -6 3 -14 -12 0=0 8-8 spl spl spl spl spl spl spl 18* 8* -2"0

.d 7-7 -14 -16 spl spl spl spl -1~ 6-6 -3 -5 -7 -9 -10 11*.~

5-5.-::: 4-4** 17 4 0 -2 -6~en 3-3 -2 -6 -9 -11 -11 spl 0

2-2 -1 -4 -6 -9 -10 spl 11


Table At (Continued)Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 ace10-7 -17 -18 -59-8 -12 -17 -410-6 -7 -8 -10 -12 -11 13 12 8 4 49-7 -8 -9 -11 -11 -11 19 13 9 0 38-8 spl spl spl spl spl spl spl sp/- sp/- h/sp

10-5 -6 -7 -9 -10 -12 11 11 7 4 49-6 -5 -6 -8 -9 -11 14 11 8 4 48-7 -4 -5 -7 -8 -10 17 18 11 4 4

"tj 10-4 -2 -4 -6 -7 -9 15 h h h 10s::~ 9-5 -3 -3 -5 -7 -9 19 h h h 10~Uj

8-6 -3 -4 -6 -7 -9 19 h h h 8-....~

:.a 7-7 h/sp h/sp spl spl spl spl 20 13 0 2"'0 10-3 1 0 -3 -4 -6 h h h h 18...~ 9-4 1 0 -2 -3 -5 h h h h 17CJI

8-5 -2 -3 -5 -6 -8 h h h h 120~ 7-6 -2 -3 -5 -6 -8 h h h h 10~

10-2 5 4 1 0 -1 h h h h h9-3 3 2 0 -1 -3 h h h h h8-4 3 1 0 -1 -3 h h h h h7-5 2 1 0 -1 -3 h h h h 206-6 h/sp h/sp h/sp h/sp h/sp h h h h h

9-2 -10 -11 -12 -13 -14 -7 -5 -3 -2 08-3 -11 -12 -13 -14 -15 -8 -5 -3 -4 -17-4 -11 -13 -14 -15 -16 -9 -5 -3 -5 -26-5 -12 -12 -14 -15 -15 -8 -5 -3 -1 -1

(l)8-2 -8 -9 -10 -11 -12 -5 -2 -1 3 3:0::s 7-3 -8 -9 -11 -11 -13 -6 -4 0 3 20

"'C 6-4 -8 -9 -11 -12 -12 -6 -3 0 5 3"'0 5-5 -8 -9 -11 -12 -14 -6 -3 -1 4 3...

Cd 7-2 0 -1 -3 -4 -5 2 7CJI

6-3 1 0 -2 -4 -4 3 80~ 5-4 1 0 -2 -4 -6 2 8~

6-2 14 9 6 3 3 145-3 14 9 5 2 1 144-4 16 10 6 2 1 145-2 17 12 9 84-3 17 11 8 7

4-2 15 11 12

3-2 15 12 13

ApPENDIX A 257

Table At (Continued)Dealer's Upcard

Player's early early & late lateHand 10 ace 7 8 9 10 ace10-7 4 16* 18 12 119-8 8 13* 16 15 1410-6 -4 -16 12 6 2 -1 -3/17*9-7 -4 -18/17* 13 6 2 -1 -3/10*8-8 -1 -18/10* 5 110-5 -3 -15 11 6 2 0 -39-6 -3 -15/19* 12 7 3 0 -1/13*8-7 -1 -14/17* 13 9 4 1 1/9*10-4 0 -13 14 13 8 4 49-5 0 -13/20* 14 13- 8 4 5/15*8-6 0 -14/19* 14 13 7 3 4/11*7-7 -5 -19/17* 8 2 -1 -3/8*10-3 5 -11 19 16 10 16

~ 9-4 5 -10/20* 19 17 11(1)

--0 8-5 1 -13/20* 19 13 11 6= 7-6 1 -14/19* 19 12 10 6(1)~~ 10-2 10 -7/20* 17~

::s 9-3 7 -10/19* 15 13~

8-4 7 -10/19* 15 137-5 7 -9/19* 16 146-6 8 -8/19* 19 156-2 -214*5-3 -2/4*4-4 0/3*

6-ace 185-2 14 -7/12* 194-3 13 -9/13* 18

5-ace 15 184-2 11 -6/15* 153-3 11 -6/15* 16

4-ace 16 193-2 12 -3/13* 17

3-ace2-2 16 -1/10*

258 PROfESSIONAL BLACKJACK

ApPENDIX A 259

Table A2High-Low, Four Decks, HI7

Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 ace

soft 18 -14 -15 -14 -15 -15 -15 h h h

hard 17 -4"'t;j hard 16 -9 -10 -11 -12 -14 9 7 5 0 3

~ hard 15 -5 -7 -8 -9 -11 10 10 8 4 5:1= hard 14 -3 -5 -6 -7 -9 17 h h h 9...t:l hard 13 0 -2 -3 -4 -7 h h h h 15

hard 12 3 1 0 -1 -3 h h h h h

11 -11 -12 -13 -14 -16 -9 -6 -4 -4 010 -8 -9 -10 -11 -13 -6 -4 -1 4 39 1 0 -2 -4 -6 3 7

0 8 13 9 5 3 1 14:c 7 16 12 9 8=0 6 20 15 11 12~

5 20 15 12 13

ace-9 10 8 6 5 4 14ace-8 8 5 3 1 0 17ace-7 0 -2 -6 -8 -13 20

0 ace-6 1 -3 -7 -10 -14:c= ace-5 15 3 -3 -6 -130 ace-4 18 6 0 -4 -10"0

ace-3 14 6 1 -1 -5ace-2 12 7 3 0 -2

ace-ace -11 -12 -13 -13 -14 -9 -8 -7 -8 -410-10 11 8 6 5 4 13 20

0 9-9 -1 -2 -3 -4 -6 6 -8 -9 2=§ 8-8 spl spl spl spl spl spl spl 17* 6* -10 7-7 -8 -10 -12 -14 -16 spl"0

0 6-6 1 -1 -3 -5 -7~ 5-5.~ 4-4** 18 12 8 7~ 3-3 8 3 0 -2 -5 13*til

2-2 7 3 0 -4 -7 spl

ace-ace -11 -12 -13 -13 -14 -9 -8 -7 -8 -40 10-10 11 8 6 5 4 13 20:c 9-9 -2 -4 -5 -6 -8 3 -8 -9 1::s0 8-8 spl spl spl spl spl spl spl 18* 8* -1~

':5 7-7 -10 -11 -13 -14 -16 spl 5/19*.~ 6-6 -2 -4 -6 -8 -10

5-5.~ 4-4** 14 6 1 -1 -6~

(/.) 3-3 0 -3 -7 -9 -11 spl 42-2 -2 -5 -7 -9 -11 spl 5


Table A2 (Continued)Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 ace10-7 -49-8 -4

10-6 -9 -10 -11 -12 -14 9 8 5 1 49-7 -9 -10 -11 -12 -14 10 8 5 0 48-8 spl spl spl spl spl spl spl sp/- sp/- h/sp

10-5 -6 -7 -8 -9 -11 10 10 8 4 59-6 -6 -6 -8 -9 -11 11 10 8 4 58-7 -5 -6 -7 -8 -11 11 11 9 4 5

~ 10-4 -3 -5 -6 -7 -9 17 h h h 10d~ 9-5 -3 -4 -6 -7 -9 18 h h h 10....,tt.:I

8-6 -3 -5 -6 -7 -9 18 h h h 9...............,:2 7-7 h/sp h/sp h/sp h/sp h/sp spl 20 h 11 8~ 10-3 0 -1 -3 -4 -6 h h h h 16......~ 9-4 0 -1 -3 -4 -6 h h h h 16(.)I

8-5 -1 -2 -4 -5 -7 h h h h 150~ 7-6 -1 -2 -4 -5 -7 h h h h 14....,

10-2 3 2 0 -1 -2 h h h h h9-3 3 1 0 -1 -3 h h h h h8-4 3 1 0 -1 -3 h h h h h7-5 2 1 0 -1 -3 h h h h h6-6 h/sp h/sp h/sp h/sp h/sp h h h h h

9-2 -11 -12 -13 -14 -15 -8 -6 -4 -4 08-3 -11 -12 -13 -14 -16 -9 -6 -4 -4 07-4 -11 -12 -13 -14 -16 -9 -6 -4 -4 06-5 -11 -12 -13 -14 -16 -9 -6 -4 -4 0

(l.)

8-2 -8 -9 -10 -11 -13 -5 -3 -1 4 3:c=' 7-3 -8 -9 -10 -11 -13 -6 -4 -1 4 30

"'C 6-4 -8 -9 -10 -11 -13 -6 -3 -1 4 3

] 5-5 -8 -9 -10 -11 -13 -6 -4 -1 4 3(.) 7-2 0 -1 -2 -4 -6 3 7I

0 6-3 1 0 -2 -4 -6 3 7~ 5-4 1 0 -2 -4 -6 3 7....,

6-2 13 9 5 3 1 145-3 13 9 5 3 1 144-4 13 9 5 2 1 145-2 16 12 9 84-3 16 11 8 8

4-2 20 15 11 123-2 20 15 12 13

ApPENDIX A 261

Table A2 (Continued)Dealer's Upcard

Player's early early &late lateHand 10 ace 7 8 9 10 ace10-7 5 18* 13 119-8 6 17* 13 1210-6 -5 -17 13 4 0 -2 -4/16*9-7 -5 -18 10 4 0 -2 -4/14*8-8 -2 -18/14* 7 010-5 -2 -14 10 6 2 0 -1/16*9-6 -2 -14 11 7 2 0 0/15*8-7 -2 -13 11 7 3 0 0/14*10-4 0 -13 16 12 7 3 4/16*9-5 0 -13 16 12 7 3 4/15*8-6 0 -13 16 12 6 3 4/14*7-7 -1 -14 11 5 2 2/13*10-3 3 -11 20 14 8

..... 9-4 4 -10 20 14 9cu"Q 8-5 3 -11 19 12 7= 7-6 2 -11 18 12 7cu.......... 10-2 8 -7 14.....::s 9-3 7 -8 13CIJ

8-4 7 -8 20 137-5 7 -8 136-6 8 -8 146-2 1/4*5-3 1/4*4-4 2/4*

6-ace 185-2 14 -6/12* 194-3 14 -7/12* 19

5-ace 15 194-2 11 -4/15* 20 163-3 11 -4/15* 20 16

4-ace 16 203-2 12 -2114* 16

3-ace2-2 15 0/11* 20

ApPENDIX A

Table A3High-Low, One Deck, S17

263

Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 acesoft 18 -17 -18 -15 -16 -17 -16 h h -1

hard 17 -19 -19 -6~ hard 16 -8 -10 -12 -13 -11 11 10 5 1 8j hard 15 -4 -6 -8 -9 -10 12 12 8 5 9rJ'J

hard 14 -2 -4 -6 -7 -8 16 h h h 13~.d hard 13 0 -1 -3 -4 -4 h h h h 20

hard 12 4 2 0 0 0 h h h h h

11 -11 -12 -13 -14 -15 -8 -5 -3 -3 010 -8 -9 -10 -11 -12 -6 -3 0 4 49 1 0 -2 -4 -5 3 8

Q.) 8 14 9 5 3 2 14:0 7 17 11 9 9=0 6 15 11 15~

5 16 12 16

ace-9 10 9 7 5 5 15ace-8 9 4 2 1 0 16ace-7 1 -1 -10 -10 -13 18

Q.) ace-6 0 -4 -8 -14 -15:0= ace-5 15 4 -3 -7 -150 ace-4 19 5 -3 -6 -10~

ace-3 13 5 0 -3 -6ace-2 11 6 1 -1 -3

ace-ace -13 -14 -15 -15 -16 -12 -10 -10 -10 -710-10 11 9 7 5 5 13 20

Q.) 9-9 0 0 -2 -4 -3 6 -14 -12 1,......0 8-8 spl spl spl spl spl spl spl 11* 6* -16=~ 7-7 -16 -17 spl spl spl spl

~6-6 1 -1 -4 -6 -7 0*5-5

.~ 4-4** 18 12 8 18~ 3-3 8 4 -1 -4 -10 13*t')

2-2 9 5 0 -4 -8 spl

ace-ace -13 -14 -15 -15 -16 -12 -10 -10 -10 -7Q.) 10-10 11 9 7 5 5 13 20:c 9-9 -1 -2 -4 -5 -5 3 -14 -12 1::s0 8-8 spl spl spl spl spl spl spl 18* 8* -16~

.s 7-7 -16 -17 spl spl spl spl -1

.~ 6-6 -2 -5 -7 -9 -11 11*5-5

.~ 4-4** 18 5 0 -2 -4~

rJ'J 3-3 -4 -7 -9 -11 -13 spl 02-2 -2 -4 -6 -9 -11 spl 11


Table A3 (Continued)Players Dealers UpcardHand 2 3 4 5 6 7 8 9 10 ace10-7 -17 -18 -79-8 -12 -17 -6


10-5 -5 -7 -8 -10 -10 11 11 7 4 99-6 -5 -6 -7 -9 -9 14 11 8 4 88-7 -4 -5 -7 -7 -8 17 18 11 4 9

~ 10-4 -1 -4 -6 -7 -7 15 h h h 15=~ 9-5 -2 -3 -5 -6 -7 19 h h h 14~

fn8-6 -2 -4 -5 -7 -7 19 h h h 12..........

~

:.E 7-7 h/sp h/sp spl spl spl spl 20 13 0 7~ 10-3 2 0 -2 -4 -3 h h h h hr-c~ 9-4 2 0 -1 -3 -3 h h h h h(,.)I

8-5 -1 -3 -4 -6 -6 h h h h 170~ 7-6 -2 -3 -5 -6 -6 h h h h 14~

10-2 6 4 2 0 0 h h h h h9-3 3 2 0 -1 -1 h h h h h8-4 3 1 0 -1 0 h h h h h7-5 2 1 0 -1 0 h h h h h6-6 h/sp h/sp h/sp h/sp h/sp h h h h h

9-2 -10 -10 -12 -13 -14 -7 -5 -3 -2 08-3 -11 -12 -13 -14 -15 -8 -5 -3 -4 07-4 -11 -13 -14 -15 -15 -9 -5 -3 -5 06-5 -11 -12 -14 -15 -15 -8 -5 -3 -1 0

Q.)

8-2 -8 -9 -9 -11 -12 -5 -2 -1 3 4:c:s 7-3 -8 -9 -11 -11 -12 -6 -4 0 3 30

"'0 6-4 -8 -9 -11 -12 -12 -6 -3 0 5 4"CS 5-5 -8 -9 -11 -12 -13 -6 -3 -1 4 4r-c~ 7-2 0 -1 -3 -4 -6 2 7(,.)I

6-3 1 0 -2 -4 -4 3 80~ 5-4 1 0 -2 -4 -6 2 8~

6-2 14 9 6 3 3 145-3 14 9 5 2 1 144-4 16 10 6 2 2 14

5-2 17 12 9 94-3 17 11 8 94-2 15 11 15

3-2 16 12 16

ApPENDIX A 265

Table A3 (Continued)Dealers Upcard

Player's early early &late lateHand 10 ace 7 8 9 10 ace10-7 4 15* 18 12 119-8 8 12* 16 15 1410-6 -4 -13 12 6 2 -1 09-7 -4 -14/20* 13 6 2 -1 0/18*8-8 -1 -12/6* 5 110-5 -3 -12 11 6 2 0 09-6 -3 -12 12 7 3 0 18-7 -1 -10/19* 13 9 4 1 3/17*10-4 0 -10 14 13 8 4 59-5 0 -9 14 13 8 4 78-6 0 -10 14 13 7 3 6/20*7-7 -5 -14/19* 8 2 -1 0/16*10-3 5 -7 19 16 10 15

.... 9-4 5 -6 19 17 11 18Q)

"t:S 8-5 1 -9 19 13 11 6 10= 7-6 1 -10 19 12 10 6 10Q).......... 10-2 10 -4/20* 17.....::s 9-3 7 -fJ/20* 15 13rn

8-4 7 -fJ/20* 15 1.37-5 7 -fJ/19* 16 146-6 8 -fJ/20* 19 156-25-34-4

6-ace 185-2 14 -4/12* 194-3 13 -5/13* 18

5-ace 15 184-2 11 -2118* 153-3 11 -2118* 16

4-aoo 16 193-2 12 0/16* 17

3-ace2-2 16 1fT*

ApPENDIX A

Table A4High-Low, Four Decks, 817

267

Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 acesoft 18 -15 -15 -15 -15 -17 -15 h h 1

hard 17 -6

1 hard 16 -8 -10 -11 -12 -13 9 7 5 0 8hard 15 -5 -6 -7 -9 -9 10 10 8 4 10

:i= hard 14 -3 -4 -6 -7 -7 17 h h h 14,d hard 13 0 -1 -3 -4 -4 h h h h 20

hard 12 3 2 0 -1 0 h h h h h

11 -11 -12 -13 -14 -15 -9 -6 -4 -4 110 -8 -9 -10 -11 -13 -6 -4 -1 4 49 1 0 -2 -4 -6 3 7

Q) 8 13 9 5 3 1 14....-4

..D 7 16 12 9 9=0 6 20 15 12 15~

5 20 15 12 16

ace-9 10 8 6 5 4 14ace-8 8 5 3 1 1 17ace-7 0 -2 -6 -8 -10 20

Q) ace-6 1 -3 -7 -10 -13:c=' ace-5 15 4 -2 -6 -120 ace-4 18 7 0 -4 -9~

ace-3 14 7 1 -1 -4ace-2 13 7 3 0 -1

ace-ace -11 -12 -12 -13 -14 -9 -8 -7 -8 -310-10 11 8 6 5 4 13 20

Q) 9-9 0 -2 -3 -4 -4 6 -8 -9 3=§ 8-8 spl spl spl spl spl spl spl 17* 6* -110 7-7 -9 -11 -13 -14 spl spl'1j

~6-6 2 0 -3 -5 -75-5

.~ 4-4** 19 13 8 18~ 3-3 8 3 a -2 -9 13*en

2-2 7 3 0 -4 -9 splace-ace -11 -12 -12 -13 -14 -9 -8 -7 -8 -3

Q) 10-10 11 8 6 5 4 13 20:c 9-9 -2 -3 -5 -6 -6 3 -8 -9 3='0 8-8 spl spl spl spl spl spl spl 18* 8* -11~

,.c:: 7-7 -9 -11 -13 -14 spl spl 5/19*~ 6-6 -1 -4 -6 -8 -10.~

5-5.-= 4-4** 15 6 1 -1 -4,.....Q.. 3-3 0 -4 -7 -9 -14 spl 4en

2-2 -3 -5 -7 -9 -12 spl 5


Table A4 (Continued)Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 ace10-7 -69-8 -510-6 -8 -9 -11 -12 -12 9 8 5 1 99-7 -8 -9 -11 -12 -12 10 8 5 0 98-8 spl spl spl spl spl spl spl sp/- sp/- h/sp10-5 -5 -6 -8 -9 -9 10 10 8 4 109-6 -5 -6 -7 -9 -9 11 10 8 4 98-7 -5 -6 -7 -8 -9 11 11 9 4 10

~ 10-4 -3 -4 -6 -7 -7 17 h h h 14=~ 9-5 -3 -4 -5 -7 -7 18 h h h 14~

til8-6 -3 -4 -5 -7 -7 18 h h h 14'-......

:E 7-7 h/sp h/sp h/sp h/sp spl spl 20 h 11 12~ 10-3 0 -1 -3 -4 -4 h h h h 'h.....~ 9-4 0 -1 -2 -4 -4 h h h h hC,,)I

8-5 0 -2 -3 -5 -5 h h h h 190~ 7-6 0 -2 -3 -5 -5 h h h h 19~

10-2 4 2 0 0 0 h h h h h9-3 3 1 0 -1 0 h h h h h8-4 3 1 0 -1 0 h h h h h7-5 3 1 0 -1 0 h h h h h6-6 h/sp h/sp h/sp h/sp h/sp h h h h h

9-2 -11 -12 -13 -14 -15 -8 -6 -4 -4 18-3 -11 -12 -13 -14 -15 -9 -6 -4 -4 17-4 -11 -12 -13 -14 -15 -9 -6 -4 -4 16-5 -11 -12 -13 -14 -15 -9 -6 -4 -4 1

Q,)8-2 -8 -9 -10 -11 -12 -5 -3 -1 4 4.....c

J:j::s 7-3 -8 -9 -10 -11 -13 -6 -4 -1 4 40"0 6-4 -8 -9 -10 -11 -12 -6 -3 -1 4 4~ 5-5 -8 -9 -10 -11 -13 -6 -4 -1 4 4a 7-2 0 -1 -2 -4 -6 3 7C,,)

I0 6-3 1 0 -2 -4 -5 3 7~ 5-4 1 0 -2 -4 -6 3 7~

6-2 13 9 5 3 1 145-3 13 9 5 3 1 144-4 13 9 5 3 1 145-2 16 12 9 94-3 16 12 9 94-2 20 15 11 153-2 20 15 12 16

ApPENDIX A 269

Table A4 (Continued)Dealer's Upcard

Player's early early &late lateHand 10 ace 7 8 9 10 ace10-7 5 17* 13 119-8 6 17* 13 1210-6 -5 -14 13 4 0 -2 -19-7 -5 -14 10 4 0 -2 08-8 -2 -14/12* 7 010-5 -2 -11 10 6 2 0 19-6 -2 -11 11 7 2 0 28-7 -2 -10 11 7 3 0 210-4 0 -9 16 12 7 3 69-5 0 -9 16 12 7 3 78-6 0 -9 16 12 6 3 67-7 -1 -10 11 5 2 510-3 3 -7 20 14 8 15

.... 9-4 4 -7 20 14 9 16(I)

"'d 8-5 3 -8 19 12 7 14= 7-6 2 -8 18 12 7 14(I)........ 10-2 8 -4 14....::s 9-3 7 -5 13tI)

8-4 7 -5 20 137-5 7 -5 136-6 8 -5 146-25-34-4

6-ace 185-2 14 -2112* 194-3 14 -3/12* 19

5-ace 15 194-2 11 -1/17* 20 163-3 11 -1/17* 20 16

4-ace 16 203-2 12 0/16* 16

3-ace2-2 15 2111* 20


Appendix BThis set of tables contains the decision rules for the

halves count (2, 7 = +0.5; 3, 4, 6 = +1; 5 = +1.5; 8 = 0; 9= -0.5; 10, ace = -1). The four tables in this part of theappendix cover both dealer actions on soft seventeen forone and four decks.


*

**

h/sp

spl

Sp/-

db

h

hI7

h/-

KEY TO APPENDIXES A AND B

Stand.

Double down.

Hit.



Either hit or split, but never stand.

Split.

Either split or stand but never hit.

number Stand (or double or split) at a count per deck equal toor greater than the number; hit (or do not split) at acount per deck less than the number.


With 4-4, splitting is better than doubling down onlyif doubling is allowed after splitting. If doubling is notallowed after splitting, then doubling on 8 is moreprofitable than splitting 4-4 no matter how high thecount. The numbers for splitting 4-4 with no doubleafter split are for use if you are not allowed to doubleon a total of eight.

ApPENDIX B

Table BlHalves, One Deck, HI7

271

Player's Dealers UpcardHand 2 3 4 5 6 7 8 9 10 acesoft 18 -18 -18 -15 -16 -15 -15 h h hhard 17 -5

"0 hard 16 -9 -10 -12 -13 -14 12 12 5 0 3

~ hard 15 -6 -6 -8 -9 -12 12 14 9 4 4+::s hard 14 -3 -3 -5 -6 -9 14 h 20 11 7:E hard 13 0 -1 -3 -4 -6 h h h h 12

hard 12 3 2 0 0 -3 h h h h h

11 -12 -12 -13 -14 -16 -8 -5 -4 -3 -110 -9 -9 -10 -11 -13 -5 -3 -1 4 39 0 0 -2 -4 -6 4 9

Cl) 8 14 9 5 3 2 17:0 7 16 11 9 9::s0 6 18 14 11 12"0

5 20 15 12 13

ace-9 9 8 6 5 4 16ace-8 8 4 2 1 0 14ace-7 0 0 -8 -8 -13 16

Cl) ace-6 0 -3 -6 -12 -14 10:0::s ace-5 15 4 -2 -6 -150 ace-4 17 4 -3 -6 -11"0

ace-3 14 5 -1 -4 -8ace-2 11 5 0 -2 -5

ace-ace -15 -14 -15 -15 -17 -12 -11 -11 -10 -810-10 11 9 6 5 5 14

Cl) 9-9 -2 -1 -3 -4 -5 5 -15 -14 -1:0::s 8-8 spl spl spl spl spl spl spl 14* 5* -20 7-7 -12 -11 -14 -15 -18 spl"0

0 6-6 0 -1 -4 -7 -8 0*~ 5-5.~ 4-4** 17 11 9 100.. 3-3 9 4 -1 -9 -6 10*en

2-2 7 4 0 -4 -6 spl

ace-ace -15 -14 -15 -15 -17 -12 -11 -11 -10 -8Cl) 10-10 11 9 6 5 5 14

:D 9-9 -3 -3 -5 -6 -7 3 -15 -14 -1::s0 8-8 spl spl spl spl spl spl spl 18* 8* -2"0

..d 7-7 -12 -11 -14 -15 -18 spl 0~ 6-6 -4 -5 -7 -10 -12 12*.~

5-5.t:: 4-4** 19 4 0 -3 -7Q..en 3-3 -7 -8 -9 -12 -13 spl -3/12*

2-2 -3 -3 -6 -8 -11 spl 8


Table HI (Continued)Players Dealer's Upcard

Hand 2 3 4 5 6 7 8 9 10 ace10-7 -59-8 -18 -5

10-6 -8 -8 -10 -12 -11 14 15 9 4 49-7 -8 -8 -11 . -11 -12 20 15 8 0 38-8 spl spl spl spl spl spl spl sp/- sp/- h/sp

10-5 -6 -6 -8 -9 -12 12 13 8 4 59-6 -6 -7 -8 -10 -12 15 12 7 3 48-7 -4 -5 -6 -7 -10 18 h 12 4 5

"'C 10-4 -2 -3 -5 -6 -9 12 h h 13 8=td 9-5 -3 -3 -5 -6 -8 17 h 20 12 8~tI)

8-6 -3 -3 -5 -6 -8 16 h 17 10 7..........~:c 7-7 h/sp h/sp spl spl spl spl 18 12 0 3"'C 10-3 0 0 -2 -3 -6 h h h h 14

""""td 9-4 0 0 -2 -3 -5 h h h h 14(,)I

8-5 -2 -2 -3 -4 -7 h h h h 100~ 7-6 -2 -2 -4 -5 -7 h h h h 9.....

10-2 5 3 1 0 -2 h h h h h9-3 2 1 -1 -2 -4 h h h h 208-4 2 0 0 -1 -4 h h h h h7-5 2 2 0 0 -2 h h h h h6-6 h/sp h/sp h/sp h/sp h/sp h h h h h

9-2 -12 -12 -13 -14 -16 -8 -6 -4 -3 -18-3 -12 -12 -13 -14 -16 -8 -5 -4 -4 -17-4 -12 -12 -13 -14 -16 -8 -5 -3 -4 -16-5 -12 -12 -13 -14 -15 -8 -5 -4 -3 0

Q,)8-2 -9 -9 -10 -11 -13 -5 -3 -3 3 2:c:s 7-3 -9 -9 -10 -11 -13 -5 -3 0 3 3

06-4 -9 -9 -10 -12 -13 -6 -3 -1 5 3""0

"0 5-5 -8 -8 -10 -11 -13 -5 -3 -1 5 4""""td 7-2 0 -1 -3 -4 -6 3 70I

6-3 1 0 -2 -4 -5 4 80~ 5-4 1 0 -2 -4 -6 3 10.....

6-2 13 8 5 3 2 165-3 15 10 6 3 1 174-4 16 9 5 3 1 16

5-2 16 12 9 94-3 16 10 9 8

4-2 18 14 11 12

3-2 20 15 12 13

ApPENDIX B 273

Table Bl (Continued)Dealer's Upcard

Player's early early & late lateHand 10 ace 7 8 9 10 ace10-7 4 17* 18 11 119-8 7 13* 14 12 1410-6 -4 -16 13 7 1 -1 -3/15*9-7 -4 -17/18* 13 6 1 -1 -2/9*8-8 -1 -14/11* 5 210-5 -3 -15 12 7 2 0 -2/17*9-6 -3 -16/19* 11 7 2 0 -1/11*8-7 -1 -13/19* 14 9 4 1 1/9*10-4 0 -12 14 11 6 3 3/18*9-5 0 -12/20* 13 11 6 4 4/13*8-6 0 -12/20* 13 11 5 3 3/11*7-7 -3 -17/19* 8 2 0 -2/9*10-3 4 -10 15 13 9 12/18*

;... 9-4 4 -10/20* 15 13 9Q)"tj 8-5 1 -12 17 12 9 6= 7-6 1 -13/20* 17 11 9 6Q);...;... 10-2 11 -9 20;...::s 9-3 7 -11/20* 13 13til

8-4 8 -11/20* 14 157-5 10 -10 17 186-6 10 -9/20* 18 186-2 -2/4*5-3 -1/5*4-4 0/3*

6-ace 195-2 15 -8/12* 204-3 14 -10/13* 19

5-ace 16 194-2 11 -6/16* 19 153-3 11 -6/16* 18 16

4-ace 17 193-2 12 -3/14* 20 17

3-ace 202-2 14 -2/11* 19

ApPENDIX B

Table B2Halves, Four Decks, HI7

275

Players Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 acesoft 18 -15 -15 -15 -15 -16 -15 h h h

hard 17 -4~ hard 16 -9 -10 -11 -13 -15 9 8 5 0 3! hard 15 -6 -7 -8 -9 -12 10 11 9 4 5CIl

hard 14 -3 -4 -5 -7 -9 14 18 19 10 8~

:E hard 13 0 -1 -3 -4 -6 h h h h 13hard 12 3 1 0 -1 -3 h h h h h

11 -12 -12 -13 -14 -16 -9 -7 -4 -4 010 -9 -9 -10 -11 -13 -6 -4 -1 4 39 1 0 -2 -4 -6 3 8

Q) 8 13 8 5 3 1 16:0 7 15 11 9 9=0 6 18 14 12 13~

5 18 15 13 14

ace-9 10 8 6 5 4 15ace-8 7 4 3 1 0 15ace-7 0 -1 -5 -7 -11 17

Q) ace-6 1 -2 -5 -9 -13 14:0= ace-5 14 3 -2 -6 -130 ace-4 17 5 0 -5 -11~

ace-3 15 6 1 -2 -6ace-2 13 7 3 0 -3

ace-ace -12 -12 -13 -14 -15 -10 -9 -8 -9 -510-10 11 8 6 5 4 14

Q) 9-9 -1 -2 -3 -4 -6 5 -9 -10 2:0::s 8-8 spl spl spl spl spl spl spl spl 5* -10 7-7 -7 -8 -10 -11 -14 spl~

0 6-6 1 0 -3 -5 -8.........

5-5~

.~ 4-4** 17 12 9 9"a 3-3 8 3 0 -2 -6 8*

CIl

2-2 6 2 0 -3 -6 spl 17

ace-ace -12 -12 -13 -14 -15 -10 -9 -8 -9 -5Q) 10-10 11 8 6 5 4 14:0 9-9 -3 -4 -5 -6 -8 3 -9 -10 1::s0 8-8 spl spl spl spl spl spl spl spl 8* -1~

.: 7-7 -12 -12 -14 -14 -17 spl 2~ 6-6 -2 -4 -6 -8 -11.~

5-5.~ 4-4** 17 6 1 -2 -7'aCIl 3-3 0 -4 -7 -10 -13 spl

2-2 -2 -4 -6 -8 -10 spl 4


Table B2 (Continued)Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 ace10-7 -49-8 -410-6 -9 -10 -11 -12 -14 10 9 6 1 49-7 -9 -10 -11 -12 -14 11 9 6 0 38-8 spl spl spl spl spl spl spl spl sp/- h/sp

10-5 -6 -7 -8 -9 -12 10 11 9 4 59-6 -6 -7 -8 -9 -12 11 11 8 4 58-7 -5 -6 -7 -9 -11 12 13 10 4 5

"t:S 10-4 -3 -4 -5 -7 -9 14 18 20 11 8=~ 9-5 -3 -4 -5 -6 -9 15 18 20 11 8~

til8-6 -3 -4 -5 -6 -9 15 19 19 10 8..........

~:a 7-7 h/sp h/sp h/sp h/sp h/sp spl 17 17 7 7"t:S 10-3 0 -1 -3 -4 -6 h h h h 13.....~ 9-4 0 -1 -3 -4 -6 h h h h 13(,,)I

8-5 -1 -2 -3 -4 -7 h h h h 120~ 7-6 -1 -2 -3 -4 -7 h h h h 12~

10-2 3 1 0 -1 -3 h h h h h9-3 2 1 0 -1 -4 h h h h h8-4 3 1 0 -1 -3 h h h h h7-5 3 1 0 -1 -3 h h h h h6-6 h/sp h/sp h/sp h/sp h/sp h h h h h

9-2 -12 -12 -13 -14 -16 -9 -7 -4 -4 08-3 -12 -12 -13 -14 -16 -9 -7 -4 -4 07-4 -12 -12 -13 -14 -16 -9 -7 -4 -4 06-5 -12 -12 -13 -14 -16 -9 -7 -4 -4 0

Q)8-2 -9 -9 -10 -11 -13 -6 -4 -1 4 3:c

::s 7-3 -9 -9 -10 -11 -13 -6 -4 -1 4 30

64 -9 -9 -10 -11 -13 -6 4 -1 4 3""0

"t:S 5-5 -8 -9 -10 -11 -13 -5 -4 -1 4 3.....~ 7-2 0 -1 -2 -4 -6 3 80I

6-3 1 0 -2 -4 -6 3 80~ 5-4 1 0 -2 -4 -6 3 8~

6-2 13 8 5 3 1 165-3 14 8 5 3 1 164-4 14 8 5 3 1 16

5-2 15 11 9 94-3 15 11 9 8

4-2 18 14 12 133-2 18 15 13 14

ApPENDIX B 277

Table B2 (Continued)Dealer's Upcard

Player's early early &late lateHand 10 ace 7 8 9 10 ace10-7 5 19* 20 11 129-8 6 18* 19 11 1210-6 -5 -17 13 5 0 -2 -4/14*9-7 -5 -17 11 5 0 -2 -4/13*8-8 -2 -18/15* 7 010-5 -2 -14 13 7 2 0 0/15*9-6 -2 -14 11 6 2 0 0/13*8-7 -2 -13 12 7 3 0 0/13*10-4 0 -11 14 10 5 3 3/15*9-5 0 -11 14 10 6 3 3/14*8-6 0 -12 14 10 5 3 3/13*7-7 0 -13 9 5 2 2/13*10-3 3 -10 16 11 7 12/15*

s-. 9-4 3 -10 16 11 8 13/14*Cl)

~ 8-5 2 -10 15 10 7 11/14*s:: 7-6 2 -11 15 10 7 11/13*Cl)s-.s-. 10-2 9 -9 17s-.::s 9-3 8 -9 15rn

8-4 8 -9 167-5 9 -9 166-6 9 -9 166-2 1/4*5-3 1/4*4-4 2/4*

6-ace 195-2 15 -7/12* 204-3 15 -8/13* 20

5-ace 16 194-2 11 -4/16* 19 163-3 11 -4/16* 18 16

4-ace 17 203-2 12 -2/15* 19 17

3-ace 202-2 14 -1/13* 19

ApPENDIX B

Table B3Halves, One Deck, S17

279

Players Dealers UpcardHand 2 3 4 5 6 7 8 9 10 acesoft 18 -18 -18 -15 -16 -17 -15 h h 0

hard 17 ~"'0 hard 16 -9 -10 -12 -13 -12 12 12 5 0 8B hard 15 -5 ~ -8 -9 -10 12 14 9 4 9(/)

hard 14 -2 -3 -5 ~ -7 14 h 20 11 12:t=..c:: hard 13 0 -1 -2 -4 -4 h h h h 17hard 12 4 2 0 0 0 h h h h h

11 -12 -12 -13 -14 -15 -8 -5 -4 -3 010 -9 -9 -10 -11 -13 -5 -3 -1 4 49 0 0 -2 -4 -5 4 9

Q) 8 13 9 5 3 2 17.,D 7 16 11 10 10::s0 6 19 14 11 15"'0

5 20 15 13 17

ace-9 9 8 6 5 4 16ace-8 8 4 2 1 0 14ace-7 0 0 -7 -8 -10 16

Q) ace-6 0 -3 -6 -12 -14 10:0=' ace-5 15 4 -2 -6 -140 ace-4 17 4 -2 -6 -10"'0

ace-3 15 5 -1 -4 -7ace-2 12 5 0 -2 -4

ace-ace -14 -14 -15 -15 -17 -12 -11 -11 -10 -710-10 11 9 6 5 5 14

Q) 9-9 -1 -1 -3 -4 -3 5 -15 -14 0:c::s 8-8 spl spl spl spl spl spl spl 14* 5* -180 7-7 -12 -12 -14 -15 sol spl"'0

0 6-6 0 -1 -4 ~ -8 0*~ 5-5.-=: 4-4** 17 12 10'a 3-3 9 4 -1 -11 -16 10*(/)

2-2 7 4 0 -4 -8 spl

ace-ace -14 -14 -15 -15 -17 -12 -11 -11 -10 -7Q) 10-10 11 9 6 5 5 14:c 9-9 -3 -3 -4 ~ ~ 3 -15 -14 0::s0 8-8 spl spl spl spl spl spl spl 18* 8* -18"'0

..c: 7-7 -12 -12 -14 -15 spl spl 0.....6~ -3 -5 -7 -10 -13 12*.~

5-5.-=: 4-4** 20 5 0 -2 -5~(/) 3-3 -8 -8 -10 -12 -16 spl -3/12*

2-2 -3 -4 ~ -8 -11 spl 8


Table B3 (Continued)Player's Dealer's Upcard

Hand 2 3 4 5 6 7 8 9 10 ace10-7 -79-8 -18 -7


10-5 -5 -6 -8 -9 -10 12 13 8 4 99-6 -6 -6 -8 -10 -10 15 12 7 3 78-7 -4 -4 -6 -7 -8 18 h 12 4 9

"0 10-4 -2 -3 -5 -6 -7 12 h h 13 12r::tU 9-5 -2 -2 -4 -6 -6 17 h 20 12 12.....CI)

8-6 -3 -3 -5 -6 -6 16 h 17 10 10""'"'.....:.a 7-7 h/sp h/sp h/sp h/sp spl spl 18 12 0 8"0 10-3 1 0 -2 -3 -3 h h h h 19~

tU 9-4 1 0 -1 -3 -3 h h h h 18(,JI

8-5 -1 -2 -3 -4 -5 h h h h 150~ 7-6 -1 -2 -3 -4 -5 h h h h 13.....

10-2 5 4 1 0 0 h h h h h9-3 2 1 0 -2 -2 h h h h h8-4 3 1 0 -1 -1 h h h h h7-5 3 2 0 0 0 h h h h h6-6 h/sp h/sp h/sp h/sp h/sp h h h h h

9-2 -12 -12 -13 -14 -15 -8 -6 -4 -3 08-3 -12 -12 -13 -14 -15 -8 -5 -4 -4 07-4 -12 -12 -13 -14 -15 -8 -5 -3 -4 06-5 -12 -12 -13 -14 -15 -8 -5 -4 -3 0

cu 8-2 -9 -9 -10 -11 -13 -5 -3 -3 3 3::c= 7-3 -8 -9 -10 -11 -12 -5 -3 0 3 40

6-4 -9 -9 -10 -12 -12 -6 -3 -1 5 4"'0

"0 5-5 -8 -8 -10 -11 -13 -5 -3 -1 5 5...tU 7-2 0 -1 -3 -4 -6 3 7(,JI

6-3 1 0 -2 -4 -5 4 80~ 5-4 1 0 -2 -4 -6 3 10.....

6-2 13 8 5 3 2 165-3 15 10 6 3 1 174-4 16 9 5 3 1 16

5-2 16 12 10 104-3 16 11 9 9

4-2 19 14 11 15

3-2 20 15 13 17

ApPENDIX B 281

Table B3 (Continued)Dealers Upcard

Player's early early &late lateHand 10 ace 7 8 9 10 ace10-7 4 17* 18 11 119-8 7 12* 14 12 1410-6 -4 -13 13 7 1 -1 09-7 -4 -14/20* 13 6 1 -1 0/17*8-8 -1 -1217* 5 210-5 -3 -12 12 7 2 0 09-6 -3 -13 11 7 2 0 0/17*8-7 -1 -10 14 9 4 1 4/17*10-4 0 -9 14 11 6 3 49-5 0 -9 13 11 6 4 68-6 0 -10 13 11 5 3 5/17*7-7 -3 -12 8 2 0 1/17*10-3 4 -7 15 13 9 12

$-c 9-4 4 -6 15 13 9 14Q)

"'0 8-5 1 -8 17 12 9 6 9= 7-6 1 -10 17 11 9 6 8/18*Q)$-c$-c 10-2 11 -5 20$-c::1 9-3 7 -7 13 13en

8-4 8 -7 14 157-5 10 -6 17 186-6 10 -7 18 186-25-34-4

6-ace 195-2 15 -4/13* 204-3 14 -5/14* 19

5-ace 16 194-2 11 -3/19* 19 153-3 11 -3/19* 18 16

4-ace 17 193-2 12 0/18* 20 17

3-ace 202-2 14 0/14* 19

282 PROFESSIONALBLACKJACK

ApPENDIX B

Table B4Halves, Four Decks, 817

283

Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 acesoft 18 -16 -16 -15 -16 -19 -15 h h 1

hard 17 -6"0 hard 16 -9 -10 -11 -13 -13 9 8 5 0 8B hard 15 -5 -6 -8 -9 -10 10 11 9 4 9C/.)

hard 14 -3 -4 -5 -6 -7 14 18 19 10 12~

:E hard 13 0 -1 -3 -4 -4 h h h h 17hard 12 3 2 0 -1 0 h h h h h

11 -12 -12 -13 -14 -16 -9 -7 -4 -4 110 -8 -9 -10 -11 -13 -6 -4 -1 4 49 0 0 -2 -4 -6 3 8

(1,) 8 13 8 5 3 1 16:0 7 15 11 9 10:s0 6 18 14 12 16'"tj

5 19 15 13 18

ace-9 10 8 6 5 4 15ace-8 7 5 3 1 1 15ace-7 0 -1 -4 -7 -8 17

Q) ace-6 1 -2 -5 -9 -12 14:c=' ace-5 15 3 -2 -6 -120 ace-4 17 6 0 -5 -10"0

ace-3 16 7 1 -1 -5ace-2 14 7 3 0 -2

ace-ace -12 -12 -13 -14 -15 -10 -9 -8 -9 -410-10 11 8 6 5 5 14

(1,) 9-9 -1 -2 -3 -4 -4 5 -9 -10 3:c:s 8-8 spl spl spl spl spl spl spl spl 5* -160 7-7 -8 -8 -10 -11 -16 spl"0

~ 6-6 1 0 -3 -5 -7~ 5-5.~ 4-4** 18 13 100.. 3-3 9 3 0 -2 -18 8*en

2-2 6 3 0 -3 -8 spl 17ace-ace -12 -12 -13 -14 -15 -10 -9 -8 -9 -4

(1,) 10-10 11 8 6 5 5 14:0 9-9 -2 -3 -4 -6 -6 3 -9 -10 3='0 8-8 spl spl spl spl spl spl spl spl 8* -16'"tj

..c 7-7 -9 -9 -11 -12 -16 spl 2....- 6-6 -2 -4 -6 -8 -11.~

5-5.~ 4-4** 18 7 1 -2 -50..

C/.) 3-3 0 -5 -8 -10 -18 spl2-2 -2 -4 -6 -8 -10 spl 4


Table B4 (Continued)Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 ace10-7 -69-8 -6

10-6 ..a -9 -11 -12 -13 10 9 6 1 89-7 ..a -9 -11 -12 -13 11 9 6 0 88-8 spl spl spl spl spl spl spl spl sp/- h/sp

10-5 -5 -6 -8 -9 -10 10 11 9 4 109-6 -5 -6 -8 -9 -10 11 11 8 4 98-7 -5 -6 -7 -8 -9 12 13 10 4 9

"t:S 10-4 -3 -4 -5 -6 -7 14 18 20 11 12=cd 9-5 -3 -4 -5 -6 -7 15 18 20 11 12~

VJ8-6 -3 -4 -5 -6 -7 15 19 19 10 12...........

~

:.a 7-7 h/sp h/sp h/sp h/sp h/sp spl 17 17 7 11"t:S 10-3 0 -1 -2 -4 -4 h h h h 17.....cd 9-4 0 -1 -2 -4 -4 h h h h 17(,)I

8-5 0 -1 -3 -4 -4 h h h h 170~ 7-6 0 -1 -3 -4 -4 h h h h 16~

10-2 4 2 0 -1 0 h h h h h9-3 3 1 0 -1 -1 h h h h h8-4 3 1 0 -1 -1 h h h h h7-5 2 2 0 -1 0 h h h h h6-6 h/sp h/sp h/sp h/sp h/sp h h h h h

9-2 -12 -12 -13 -14 -16 -9 -7 -4 -4 18-3 -11 -12 -13 -14 -16 -9 -7 -4 -4 17-4 -11 -12 -13 -14 -16 -9 -7 -4 -4 16-5 -12 -12 -13 -14 -16 -9 -7 -4 -4 1

Q)8-2 -9 -9 -10 -11 -13 -6 -4 -1 4 4:0

::s 7-3 ..a -9 -10 -11 -13 -6 -4 -1 4 40

64 -8 -9 -10 -11 -13 -6 4 -1 4 4"'t:S

"'0 5-5 -8 -9 -10 -11 -13 -5 -4 -1 4 4.....cd 7-2 0 -1 -2 -4 -6 3 8(,)I

6-3 1 0 -2 -4 -6 3 80~ 5-4 1 0 -2 -4 -6 3 8~

6-2 13 8 5 3 1 165-3 14 8 5 3 1 164-4 14 8 5 3 1 16

5-2 15 11 9 104-3 15 11 9 10

4-2 18 14 12 16

3-2 19 15 13 18

ApPENDIX B 285

Table B4 (Continued)Dealer's Upcard

Player's early early &late lateHand 10 ace 7 8 9 10 ace10-7 5 18* 20 11 129-8 6 17* 19 11 1210-6 -5 -14 13 5 0 -2 -19-7 -5 -14 11 5 0 -2 08-8 -2 -13/12* 7 010-5 -2 -11 13 7 2 0 19-6 -2 -11 11 6 2 0 18-7 -2 -10 12 7 3 0 210-4 0 -9 14 10 5 3 59-5 0 -8 14 10 6 3 68-6 0 -9 14 10 5 3 57-7 0 -9 9 5 2 410-3 3 -7 16 11 7 12

10-4 9-4 3 -7 16 11 8 12Q.)

"'0 8-5 2 -7 15 10 7 11c: 7-6 2 -7 15 10 7 11Q.)10-410-4 10-2 9 -5 1710-4

=' 9-3 8 -5 15en

8-4 8 -5 167-5 9 -5 166-6 9 -5 166-25-34-4

6-ace 195-2 15 -3/13* 204-3 15 -3/13* 20

5-ace 16 194-2 11 -1/18* 19 163-3 11 -1/18* 18 16

4-ace 17 203-2 12 0/18* 19 17

3-ace 202-2 14 1/15* 19


APPENDIXC

This appendix lists some simulation results generated by Blackjack Count Analyzer. The tables in thisappendix show the frequencies with which certain countsper deck appear, and the win rates at those counts.

Each count per deck is truncated before beinginserted into a table. For example, a hand with a countper deck of +4.8 appears as part of the +4 group, and ahand with a count per deck of -4.8 appears as part of the-4 group.

There are three zero groups. The columns labeled 0mean exactly zero. The columns labeled +0 mean greaterthan zero but less than +1. The columns labeled -0 meannegative but greater than -1. For single deck there is no-0 or +0 because it is impossible to get such a count perdeck with the high-low system.

Each table in this appendix contains three rows. ThefIrst row is the frequency with which each count per deckoccurred in the simulation. The second row is the winrate for each count per deck as determined by thesimulation. The third row is the standard error appli-

ApPENDIX C 287

cable to the win rate. The win rates are estimates, andthe standard errors define the precision of those estimates.

Smaller standard errors are due to larger samplesizes. The win rates for the one, two, and six-deckbenchmarks are based on sample sizes exceeding 600million, so those standard errors are small. The otherwin rates are based on smaller samples - 100 millionor more - and thus have larger standard errors.

For a given count per deck, the win rate is fairlyconstant from table to table - if the same rules andcounting system are used. For the benchmark rules,each count per deck is worth about 0.5% up to thethreshold for buying insurance, above which it is worthmore than 0.5%. For double exposure, table C7 showseach count per deck to be worth about 0.7%.

Penetration is 52 cards for tables C1, C2, C6, C7, andC8. It is 26 cards for C3, 62 cards for C4, and 42 cards forC5.

For a given set of rules, the different win ratesassociated with different numbers of decks and different levels of penetration are due to the frequencies withwhich high counts occur. For example, for the six-deckbenchmark, counts per deck of +6 and over occur 2.49%of the time. With 62-card penetration, counts per deck of+6 and above occur only 2.04% of the time.

To generate more such tables as are shown in thisappendix, get Blackjack Count Analyzer.


Table ClFrequencies and Win Rates

6-Deck BenchmarkCount per deck

-6& +6&under -5 -4 -3 -2 -1 -0 0 +0 +1 +2 +3 +4 +5 over

f % 2.43 1.51 2.38 4.04 6.69 11.56 17.18 9.24 16.77 11.32 6.48 4.03 2.37 1.51 2.49

ev 95.77 96.68 97.08 97.61 98.21 98.67 99.09 99.38 99.70 100.22100.76101.35101.91102.50104.21

se .03.04 .03 .03 .02 .02 .01 .02 .01 .02 .02 .03 .04 .05 .04

Table C2Frequencies and Win Rates

2-Deck BenchmarkCount per deck

~& ~&

under -5 -4 -3 -2 -1 -0 0 +0 +1 +2 +3 +4 +5 overf% 2.20 1.75 2.78 4.44 7.42 11.64 9.17 23.82 9.06 10.56 6.41 4.42 2.51 1.65 2.18

ev 95.91 96.78 97.24 97.75 98.24 98.76 99.22 99.53 99.95 100.38101.02101.61102.18102.67104.10

se .03 .04 .03 .02 .02 .02 .02 .01 .02 .02 .02 .03 .03 .04 .04


I-Deck BenchmarkCount per deck

-6 & +6&under -5 -4 -3 -2 -1 -0 0 +0 +1 +2 +3 +4 +5 over

f% 5.52 3.67 3.37 5.12 5.69 10.12 0 36.63 0 9.80 4.73 4.82 2.87 2.03 5.64

ev 95.94 97.11 97.50 98.02 98.48 99.18 99.81 100.84101.43102.18102.68103.49105.09

se .02 .03 .03 .02 .02 .02 .01 .02 .02 .02 .03 .04 .02

ApPENDIX C 289


6-Deck, Shuffle at 62Count per deck

~& ~&

under -5 -4 -3 -2 -1 -0 0 +0 +1 +2 +3 +4 +5 overf% 2.00 1.40 2.30 3.99 6.72 11.76 17.60 9.35 17.15 11.50 6.53 3.96 2.30 1.39 2.03

ev 95.73 96.68 97.15 97.68 98.18 98.66 99.09 99.41 99.69 100.19100.64101.37101.93102.49104.05

sa .07 .08 .07 .05 .04 .03 .02 .03 .03 .03 .04 .05 .07 .09 .08


6-Deck, Shuffle at 42Count per deck

-6& +6&under -5 -4 -3 -2 -1 -0 0 +0 +1 +2 +3 +4 +5 over

f% 2.94 1.59 2.49 4.13 6.70 11.40 16.56 9.16 16.15 11.18 6.49 4.11 2.50 1.60 3.02

ev 95.80 96.83 97.19 97.75 98.10 98.64 99.14 99.23 99.73 100.21100.93101.33101.97102.59104.41sa .10 .14 .11 .09 .07 .05 .04 .05 .05 .06 .07 .09 .12 .15 .11


Halves, 6-DeckCount per deck

~& ~&

under -5 -4 -3 -2 -1 -0 0 +0 +1 +2 +3 +4 +5 overf % 2.78 1.62 2.54 4.14 6.76 11.48 17.96 6.05 17.60 11.24 6.63 4.16 2.56 1.63 2.85ev 95.81 96.78 97.20 97.75 98.14 98.65 99.09 99.40 99.68 100.15100.72101.29101.83102.35104.19

se .05 .06 .05 .04 .03 .02 .02 .03 .02 .02 .03 .04 .05 .07 .05



Double Exposure BenchmarkCount per deck

-6& +6&under -5 -4 -3 -2 -1 -0 0 +0 +1 +2 +3 +4 +5 over

f% 2.46 1.52 2.40 4.02 6.70 11.64 16.98 9.41 16.72 11.40 6.48 3.98 2.31 1.51 2.47

ev 94.43 96.20 96.93 97.55 98.24 98.94 99.45 99.79 100.09100.72101.38102.05102.58103.24104.16se .06 .08 .06 .05 .04 .03 .02 .03 .02 .03 .04 .05 .06 .08 .06


Double Exposure HalvesCount per deck

-6& +6&under -5 -4 -3 -2 -1 -0 0 +0 +1 +2 +3 +4 +5 over

f% 2.83 1.64 2.59 4.22 6.76 11.46 17.95 6.23 17.41 11.25 6.59 4.13 2.51 1.62 2.81

ev 94.58 96.35 96.98 97.76 98.18 98.82 99.47 99.78 100.11100.72101.34102.08102.54 103.29104.90

se .09 .12 .10 .08 .06 .05 .04 .07 .04 .05 .06 .08 .10 .13 .10

291

APPENDIX D

This appendix contains simulation data showinghow often you are likely to face each situation. There aretwo ways to do it. One way is to look at only the initialhands. Another way is to look at how often you encountereach situation in a casino; that is, recognize that youmight have to make more than one decision on one hand.For example, if you are dealt 4-3 and hit with 7, you haveto make a decision with a total of seven and also withfourteen, and if you hit your fourteen you may have tomake yet another decision before you are done with thehand.

I am presenting the data both ways. Tables D1 andD3 recognize only the decision you had to make withyour initial total of seven, and tables D2 and D4 recognize all the decisions you had to make on the hand.

Tables D1 and D2 use one deck, and tables D3 andD4 use six decks. All four tables use benchmark rulesand basic strategy. There is no double counting; forexample, 4-4 shows up as a pair of 4s only, and is notincluded in the number for hand total of eight.


The dealer is assumed to check the hole card, so alldecisions against a 10 and ace are based on the dealernot having a natural. If the dealer has a natural, theplayer hand totals are not added in to any of the tablesin this appendix.

After you double down and receive another card youdo not have a decision to make, so your total afterdoubling down is not included in tables D2 and D4.Likewise with your final hands after splitting a pair ofaces.

The data in the tables in this appendix are expressed as frequencies per 100,000 hands. They aresimulation results from a modified version of BlackjackCount Analyzer, for samples of more than 100 millionhands each. The simulations played two hands at a timeof basic strategy with benchmark rules, including shuffling if 52 or fewer cards remain for the six-deck numbers, and if 26 or fewer cards remain for the single deck.Use of different rules, different numbers of simultaneous hands, or different shuflle points could get slightlydifferent numbers for tables D2 and D4.

Individual frequencies might be slightly different(higher or lower by 1 per 100,000) for other 100-millionhand samples. The amount of sampling error can beseen by comparing the numbers in tables D 1 and D2that ought to be the same. For example, player 10-10against 2: table Dl says 723 and table 02 says 724. Thatdifference is due solely to the two tables being generatedfrom different 100-million-hand samples.

Most of the deviation of those numbers from 36 isdue to having a fixed shuffle point rather than dealinga constant number of rounds between shuffles. Dealinganother round is more likely from a pack poor in facecards. This effect is much more pronounced in single

ApPENDIX D 293

deck than in multiple-deck games, and is easy to seefrom the frequencies of 10-10 in table Dl. You mightexpect each frequency against small upcards to be(16xI5x4)/(52x51x50) or 724 per 100,000 hands, but theactual frequencies range downward from 724.

The only hands missing from table D1: Dealernaturals were 4808 per 100,000, of which 176 tied playernaturals. Dealer naturals were equally split between 10up and ace up.

The only hands missing from table D3: Dealernaturals were 4744 per 100,000, of which 225 tied playernaturals. Dealer naturals were equally split between 10up and ace up.

Lawrence Revere presents first-two-card data forone deck on page 7] of Playing Blackjack as a Business.Julian Braun presents first-twa-card data for four deckson page 82 of How to Play Winning Blac/fjack. Chambliss and Roginski present first-twa-card data for sixdecks on page 42 of Playing Blackjack in Atlantic City.All three of these data sets apply only if a fixed numberof rounds are dealt between shuftles. For example if youface a shuffle after every round, the data of Revere,Braun, and Chambliss/Roginski describe the frequencywith which each situation arises.



Table DlFrequencies (per 100,000) of

Initial Hands, 1 Deck, S17Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 AA-A 36 36 36 36 36 36 36 36 139 1210-10 723 722 720 720 717 716 716 714 2291 5149-9 36 36 36 36 36 36 36 18 131 248-8 36 36 36 36 36 36 18 36 131 247-7 36 36 36 36 36 18 36 36 132 256-6 36 37 36 36 18 36 36 36 133 255-5 37 37 37 18 36 36 36 36 133 254-4 37 37 18 37 37 36 37 37 134 253-3 37 19 37 37 37 37 37 36 134 252-2 19 37 37 37 37 37 37 37 135 25aoo-10 388 386 387 386 384 384 384 384 1347 200ace-9 97 97 97 97 96 96 96 72 360 49ace-8 97 97 97 97 97 97 72 96 360 49ace-7 98 97 97 96 97 72 97 96 361 49ace-6 98 97 97 97 73 97 97 97 362 49ace-5 98 97 97 73 97 97 97 96 362 49ace-4 98 98 73 97 97 97 97 97 362 49ace-3 98 73 98 97 97 97 97 97 363 49ace-2 74 98 98 98 98 98 98 98 364 50hard 19 386 385 385 384 383 383 382 286 1313 266hard 18 386 387 386 384 384 383 286 382 1312 267hard 17 485 482 482 482 481 384 455 455 1666 332hard 16 485 484 482 482 385 456 479 455 1667 332hard 15 583 582 581 483 555 554 553 551 2024 398hard 14 585 583 486 557 556 578 553 553 2025 400hard 13 683 585 656 655 655 651 651 651 2387 465hard 12 586 658 656 656 679 654 652 651 2387 46511 368 367 365 366 365 364 364 362 1416 26210 270 269 269 293 268 267 266 291 1064 1979 271 270 269 269 268 268 292 291 1067 1988 172 171 197 171 171 195 195 194 711 1327 173 173 172 172 196 196 195 195 715 1326 74 99 74 98 98 98 98 98 358 665 74 74 99 99 98 98 98 97 358 66

ApPENDIX D 295

Table D2Frequencies (per 100,000) of

Decisions, 1 Deck, 817Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 AA-A 37 36 37 36 36 36 36 36 139 1210-10 724 722 721 719 717 715 715 714 2292 5139-9 39 40 39 39 39 36 39 19 131 248-8 39 39 39 39 39 39 19 39 143 267-7 39 39 39 39 39 19 36 36 132 246-6 37 40 40 40 19 36 36 36 133 245-5 37 37 37 18 37 36 36 36 133 254-4 37 37 18 37 37 37 36 37 133 253-3 37 19 40 40 40 40 37 37 134 252-2 19 37 40 40 40 40 37 37 135 25BJ 388 387 387 386 385 383 383 383 1347 201soft 21 not BJ 46 40 29 9 9 50 47 62 334 44soft 20 150 143 130 111 110 168 176 160 681 91soft 19 161 155 146 125 125 169 140 177 663 87soft 18 157 152 139 122 123 138 158 157 587 78soft 17 133 142 126 114 86 137 139 139 520 68soft 16 122 123 125 83 107 131 128 128 479 64soft 15 109 112 86 98 98 110 112 112 420 55soft 14 103 79 111 105 105 110 103 103 386 52soft 13 74 98 105 105 105 104 97 97 364 49hard 21 79 85 28 29 29 371 365 369 1445 362hard 20 80 86 31 32 33 360 355 374 1777 363hard 19 488 495 440 442 439 835 877 772 3158 640hard 18 554 562 508 506 503 876 763 919 3401 678hard 17 712 688 628 624 628 927 976 993 3773 729hard 16 687 713 611 610 502 894 924 910 3464 666hard 15 793 784 708 607 675 942 934 950 3599 688hard 14 764 776 610 645 640 868 853 867 3291 628hard 13 879 795 818 772 769 912 884 899 3390 645hard 12 706 774 789 771 799 827 797 810 3059 59111 432 439 452 454 451 462 453 451 1731 31910 321 330 343 370 345 337 323 350 1282 2379 300 310 322 327 322 316 327 326 1196 2208 185 192 228 203 198 221 213 213 780 1447 178 180 1~O 190 216 216 204 205 747 1386 75 100 79 105 105 105 100 99 364 675 75 74 112 112 112 112 98 98 359 67


Table D3Frequencies (per 100,000) ofInitial Hands, 6 Decks, S17

Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 AA-A 44 44 44 44 44 44 44 44 164 2810-10 727 728 727 726 725 726 726 726 2619 5069-9 44 44 44 44 44 44 44 40 162 308-8 44 44 44 44 44 44 41 44 162 317-7 44 44 44 44 44 40 44 44 162 306-6 44 44 44 44 40 44 44 44 162 315-5 44 44 44 41 44 44 44 44 162 304-4 44 44 41 44 44 44 44 44 162 313-3 44 41 44 44 44 44 45 44 163 312-2 41 44 44 44 44 44 44 44 163 31ace-10 368 367 367 368 368 368 367 366 1345 244ace-9 92 92 92 92 93 92 92 88 340 61ace-8 92 92 92 92 92 92 88 92 339 60ace-7 92 92 92 93 92 88 92 92 342 61ace-6 92 92 92 92 89 92 92 92 341 61ace-5 92 92 92 88 92 92 92 93 341 61ace-4 92 92 88 92 92 92 92 92 341 61ace-3 92 89 92 92 92 92 92 92 341 61ace-2 88 93 92 92 92 92 92 91 341 61hard 19 368 367 369 368 366 367 366 351 1338 254hard 18 367 368 366 368 368 366 351 367 1340 254hard 17 460 461 459 460 459 444 455 455 1679 319hard 16 460 460 461 459 444 454 459 455 1677 318hard 15 552 553 552 536 547 548 547 548 2016 382hard 14 551 552 537 548 547 551 547 548 2018 383hard 13 644 630 641 639 640 639 640 638 2356 446hard 12 629 640 641 641 643 640 639 640 2360 44611 365 364 365 365 364 365 364 363 1358 25410 273 273 273 277 272 273 272 277 1016 1909 272 273 272 272 272 272 277 276 1017 1908 181 181 185 181 180 184 185 184 680 1277 181 180 181 181 184 184 185 185 680 1276 89 92 89 92 92 92 92 92 340 645 89 89 92 92 92 93 93 92 340 63

ApPENDIX D 297

Table D4Frequencies (per 100,000) of

Decisions, 6 Decks, S17Player's Dealer's UpcardHand 2 3 4 5 6 7 8 9 10 AA-A 44 44 44 44 44 44 44 44 164 2810-10 727 727 728 726 726 725 725 727 2620 5059-9 51 51 51 51 51 44 51 47 162 308-8 51 51 52 51 51 51 47 51 188 357-7 52 51 51 51 51 47 44 44 162 316-6 44 52 51 52 47 44 44 44 162 305-5 44 44 44 40 44 44 44 44 162 314-4 44 44 40 44 44 44 44 44 163 303-3 44 41 52 51 51 51 44 44 163 302-2 41 44 51 52 52 52 44 44 163 31BJ 368 367 368 368 368 368 367 366 1345 244soft 21 not BJ 48 42 29 20 10 50 48 59 312 55soft 20 168 141 126 118 108 161 168 174 633 113soft 19 161 155 142 133 124 164 157 173 640 114soft 18 156 150 137 128 119 155 151 150 554 99soft 17 134 142 126 119 105 133 135 135 499 89soft 16 121 122 118 106 102 125 122 122 451 80soft 15 109 110 103 100 93 108 110 110 406 72soft 14 99 95 108 108 101 107 98 99 367 65soft 13 88 92 100 100 100 100 92 92 340 61hard 21 95 87 32 32 32 377 369 373 1408 351hard 20 93 91 36 36 37 376 365 373 1762 358hard 19 575 487 435 434 434 826 850 836 3130 620hard 18 577 555 501 501 501 863 835 902 3431 674hard 17 728 678 622 620 616 991 969 985 3726 727hard 16 700 702 604 601 581 894 900 911 3446 671hard 15 802 779 690 672 678 937 927 941 3551 688hard 14 789 769 672 652 647 850 853 867 3269 634hard 13 887 851 820 810 780 914 873 886 3337 646hard 12 787 772 797 788 783 831 789 801 3018 58511 464 453 469 468 468 473 459 457 1677 31410 341 351 363 367 364 355 334 338 1250 2339 315 323 335 336 335 326 313 311 1150 2168 201 209 224 220 219 216 205 205 755 1417 190 190 205 205 209 208 195 195 717 1346 91 96 96 100 100 100 95 95 352 665 89 89 109 108 108 108 92 92 340 64


APPENDIX E

Over the years I have made considerable use of theappendix tables in Edward O. Thorp's Beat the Dealer. Iremember the first time I used them: I had just learnedabout a casino offering surrender, and I wanted to knowwhich player totals have an expectation of worse than-50%. I am reminded of this every time I use those tablesbecause I underlined all player totals of fifteen andsixteen against the dealer's 10 in my copy of Beat theDealer.

The purpose of this appendix is to provide a morecomplete version of that part of the appendix of Beat theDealer. Thorp's tables give expected values for singledeck, dealer stands on soft seventeen, player can doubledown after splitting. My tables give expected values forthat, and also six decks, dealer hits soft seventeen, andno doubling down after splitting.

Each dealer upcard has a separate page. If thedealer upcard could turn into a soft seventeen, there areseparate pages for the two dealer actions on soft seventeen. (What the dealer does on soft seventeen does not

ApPENDIX E 299

matter for upcards of 7, 8, 9, and 10.) The first set of tablesis for single deck; and then comes a set of tables for sixdecks.

Each table has· four parts, one each for standing,hitting, doubling down, and splitting. For the two-cardhand in which you are interested, find the row corresponding to one of your cards and the column corresponding to the other. (For pair splits you need find onlythe column for one of your cards.)

Here is an example using the tables. Find the tablefor single deck, dealer stands on soft seventeen andshows a 2. Look at the various ways you can play 4-4against 2. For standing you should find -.291. Thatmeans if you are so foolish as to stand on 4-4 against 2,on average you will lose 29.1% of your bet. You will notlose it 100% of the time because if the dealer busts youwin. For hitting 4-4 against 2 you should fmd -.014. Fordoubling down on 4-4 against 2 you should find -.186.The value for doubling down is less than twice the valueof hitting since if 4-4 is hit with a small enough card youwould hit again. The -.186 for doubling down is expressed as a fraction of your initial bet. That is, doublingdown 4-4 against 2 costs you 18.6% of your initial bet,which is 9.3% of your doubled bet. The expectation ofsplitting 4-4 against 2 depends on whether you candouble down after splitting; it is -.238 if you are notallowed to double down after splitting, and -.152 if youare allowed to double after splitting. As with the doubledown numbers, the expectations for splitting are expressed in units of original bets. Thus losing 16.0% of theoriginal bet means losing 8.0% on each of the two splithands. For the example of 4-4 against 2 the highestexpected value is the -.014 for hitting, which is why basicstrategy recommends hitting 4-4 against 2.


To calculate such tables, certain assumptions andapproximations must be made, and different assumptions or approximations result in different numbers.The method of calculation I use is to remove three cardsmaximum for the dealer and three cards maximum forthe player (three cards per hand on splits), and thensample with replacement from the remaining pack forany other cards needed to complete the player or dealerhand. Sampling with replacement is the same thing asassuming an infinite number of decks, each of which ismissing the cards I have removed. Specifying threedealer cards and three player cards is more accuratethan simply assuming each denomination has 1/13chance of popping up as the next card, but requiresconsiderably more calculation time. Removing a fourthcard for the dealer and a fourth card for the player wouldtake about ten times as much calculation time (not 100times as much because most hands do not require afourth card), and would result in a small increase inaccuracy.

Thorp's 1966 numbers are more precise than mine,but only slightly so. The main difference is on splits:Thorp assumes resplits to four hands, and I assume noresplits. If splitting is the best way to play a pair, thetables will say so no matter whether the calculationsassume no resplits, unlimited resplits, or something inbetween. If splitting is recommended, then basic strategy will also say to resplit. Therefore, the tables yieldgood advice on splits and resplits no matter how thecalculations on resplits are done.

Another difference between Thorp's numbers andmine is I use total-dependent basic strategy to completethe player hands, and Thorp uses a mixture of totaldependent and composition-dependent. For example,

ApPENDIX E 301

against a dealer's 3, Thorp uses a hard standing nwnberof twelve for analyzing hands starting with 7-2, but ahard standing number of thirteen for hands startingwith 6-2, 6-3, 7-3, 8-2, or 8-3. That means he stands on 72-3 if the 3 is the hit card, but hits if 2 is the hit card. Ithink that standing on 7-2-3 but hitting 7-3-2 is silly; Ihit all twelves against a 3.

If you compare this appendix with other parts of thisbook, you might fmd small inconsistencies. Those inconsistencies are unimportant for how fast you can winfrom casinos. This appendix has been calculated usinga more time-conswning but more accurate method thanwas used for the main part of the book and appendixesA and B.

Probabilities for dealer showing an ace or 10 areconditional on the dealer not having a natural.

For player naturals, the payoff is 1.5 units in mostcasinos. However, the tables in this appendix do notshow 1.500 for those hands. What the tables show forplayer hands of A-I0 are the expected values of playingthe hand as an ordinary twenty-one. The expectedvalues are less than 1.000 to the extent the dealer alsogets to twenty-one in any number of cards. Though thesevalues for A-I0 have no meaning in casinos wherenaturals win automatically, substituting 1.500 into thetables in those places would have meant eliminatingnonobvious numbers that you conceivably might want.

If you are fascinated by how the calculations aredone and want to delve further into it, I suggest TheTheory of Blaclg·ack by Peter A. Griffin.



1 Deck, S17, Dealer Shows AceEV Stand

2 3 4 5 6 7 8 9 10 AceAce -.662 -.662 -.659 -.660 -.483 -.101 .290 .681 .925 -.66410 -.673 -.673 -.671 -.670 -.654 -.467 -.082 .308 .650 .9259 -.658 -.657 -.655 -.654 -.637 -.644 -.452 -.055 .308 .6818 -.652 -.651 -.649 -.648 -.631 -.638 -.644 -.452 -.082 .2907 -.646 -.645 -.642 -.641 -.624 -.631 -.638 -.644 -.467 -.1016 -.640 -.639 -.637 -.635 -.618 -.624 -.631 -.637 -.654 -.4835 -.658 -.657 -.655 -.654 -.635 -.641 -.648 -.654 -.670 -.6604 -.659 -.659 -.656 -.655 -.637 -.642 -.649 -.655 -.671 -.6593 -.660 -.661 -.659 -.657 -.639 -.645 -.651 -.657 -.673 -.6622 -.661 -.660 -.659 -.658 -.640 -.646 -.652 -.658 -.673 -.662

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace -.068 -.101 -.154 -.206 -.200 -.109 -.017 .067 .152 -.03110 -.349 -.393 -.445 -.499 -.509 -.556 -.633 -.743 -.883 .1529 .170 -.375 -.383 -.441 -.496 -.495 -.545 -.625 -.743 .0678 .086 .171 -.375 -.433 -.453 -.456 -.495 -.545 -.633 -.0177 -.083 .087 .171 -.377 -.444 -.495 -.456 -.495 -.556 -.1096 -.226 -.078 .081 .173 -.386 -.444 -.453 -.496 -.509 -.2005 -.331 -.217 -.070 .091 .173 -.377 -.433 -.441 -.499 -.2064 -.334 -.345 -.209 -.070 .081 .171 -.375 -.383 -.445 -.1543 -.291 -.333 -.345 -.217 -.078 .087 .171 -.375 -.393 -.1012 -.258 -.291 -.334 -.331 -.226 -.083 .086 .170 -.349 -.068

EV Double Down2 3 4 5 6 7 8 9 10 Ace

Ace -.594 -.597 -.621 -.657 -.528 -.358 -.193 -.041 .145 -.60010 -.811 -.859 -.932 -1.013 -1.018 -1.112 -1.266 -1.485 -1.766 .1459 .189 -.859 -.838 -.921 -1.008 -.991 -1.089 -1.250 -1.485 -.0418 .025 .205 -.847 -.931 -.942 -.928 -.990 -1.089 -1.266 -.1937 -.403 .040 .222 -.838 -.937 -1.030 -.928 -.991 -1.112 -.3586 -.800 -.402 .048 .240 -.835 -.937 -.942 -1.008 -1.018 -.5285 -1.126 -.795 -.396 .055 .240 -.838 -.931 -.921 -1.013 -.6574 -1.297 -1.127 -.797 -.396 .048 .222 -.847 -.838 -.932 -.6213 -1.320 -1.299 -1.127 -.795 -.402 .040 .205 -.859 -.859 -.5972 -1.321 -1.320 -1.297 -1.126 -.800 -.403 .025 .189 -.811 -.594

EV Split2 3 4 5 6 7 8 9 10 Ace

reg -.428 -.471 -.547 -.648 -.638 -.607 -.340 -.072 .098 .224dbs -.428 -.471 -.547 -.648 -.638 -.607 -.340 -.072 .098 .224

ApPENDIX E 303

I Deck, HI7, Dealer Shows AceEV Stand

2 3 4 S 6 7 8 9 10 Ace

Ace -.590 -.588 -.586 -.588 -.508 -.224 .197 .625 .907 -.59210 -.602 -.600 -.598 -.599 -.595 -.505 -.212 .213 .594 .9079 -.588 -.586 -.584 -.585 -.580 -.575 -.492 -.187 .213 .6258 -.582 -.579 -.577 -.578 -.573 -.568 -.575 -.492 -.212 .1977 -.575 -.573 -.570 -.571 -.566 -.561 -.568 -.575 -.505 -.2246 -.581 -.579 -.577 -.576 -.571 -.566 -.573 -.580 -.595 -.5085 -.586 -.585 -.582 -.584 -.576 -.571 -.578 -.585 -.599 -.5884 -.585 -.583 -.581 -.582 -.577 -.570 -.577 -.584 -.598 -.5863 -.586 -.585 -.583 -.585 -.579 -.573 -.579 -.586 -.600 -.5882 -.588 -.586 -.585 -.586 -.581 -.575 -.582 -.588 -.602 -.590

EV Hit2 3 4 S 6 7 8 9 10 Ace

Ace -.108 -.140 -.189 -.243 -.230 -.174 -.077 .018 .112 -.07110 -.380 -.421 -.469 -.524 -.529 -.577 -.645 -.748 -.884 .1129 .130 -.405 -.411 -.468 -.518 -.521 -.566 -.637 -.748 .0188 .038 .133 -.404 -.461 -.477 -.483 -.520 -.566 -.645 -.0777 -.142 .039 .136 -.407 -.467 -.524 -.483 -.521 -.577 -.1746 -.279 -.125 .044 .141 -.405 -.467 -.477 -.518 -.529 -.2305 -.366 -.280 -.125 .042 .141 -.407 -.461 -.468 -.524 -.2434 -.363 -.377 -.268 -.125 .044 .136 -.404 -.411 -.469 -.1893 -.323 -.363 -.377 -.280 -.125 .039 .133 -.405 -.421 -.1402 -.294 -.323 -.363 -.366 -.279 -.142 .038 .130 -.380 -.108


Ace -.553 -.550 -.571 -.609 -.531 -.416 -.233 -.064 .136 -.56010 -.818 -.877 -.952 -1.052 -1.058 -1.154 -1.290 -1.495 -1.769 .1369 .181 -.865 -.851 -.948 -1.043 -1.042 -1.132 -1.274 -1.495 -.0648 .002 .202 -.850 -.944 -.966 -.972 -1.040 -1.132 -1.290 -.2237 -.434 .017 .223 -.839 -.948 -1.060 -.972 -1.042 -1.154 -.4166 -.819 -.424 .030 .233 -.833 -.948 -.966 -1.043 -1.058 -.5315 -1.075 -.815 -.420 .026 .233 -.839 -.944 -.948 -1.052 -.6094 -1.160 -1.069 -.810 -.420 .030 .223 -.850 -.851 -.952 -.5713 -1.172 -1.160 -1.069 -.815 -.424 .017 .202 -.865 -.877 -.5502 -1.175 -1.172 -1.160 -1.075 -.819 -.434 .002 .181 -.818 -.553

EV Split2 3 4 5 6 7 8 9 10 Ace

reg -.502 -.542 -.605 -.716 -.684 -.687 -.484 -.193 -.004 .214dbs -.491 -.528 -.587 -.697 -.664 -.669 -.471 -.186 -.004 .214


1 Deck, S17, Dealer Shows 2EV Stand

2 3 4 S 6 7 8 9 10 Ace

Ace -.284 -.285 -.285 -.268 -.133 .135 .401 .656 .880 -.27610 -.312 -.314 -.312 -.296 -.299 -.159 .118 .385 .627 .8809 -.288 -.289 -.287 -.271 -.275 -.278 -.138 .137 .385 .6568 -.283 -.285 -.283 -.266 -.269 -.274 -.276 -.138 .118 .4017 -.282 -.282 -.280 -.264 -.267 -.270 -.274 -.278 -.159 .1356 -.278 -.280 -.277 -.260 -.263 -.267 -.269 -.275 -.299 -.1335 -.275 -.277 -.275 -.257 -.260 -.264 -.266 -.271 -.296 -.2684 -.292 -.294 -.291 -.275 -.277 -.280 -.283 -.287 -.312 -.2853 -.294 -.296 -294 -.277 -.280 -.282 -.285 -.289 -.314 -.2852 -.292 -.294 -.292 -.275 -.278 -.282 -.283 -.288 -.312 -.284

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .039 .016 -.013 -.033 .006 .064 .119 .190 .246 .09410 -.244 -.305 -.369 -.436 -.466 -.539 -.633 -.750 -.847 .2469 .263 -.267 -.293 -.359 -.430 -.456 -.530 -.628 -.750 .1908 .212 .267 -.260 -.332 -.363 -.391 -.454 -.530 -.633 .1197 .082 .214 .272 -.253 -.332 -.407 -.391 -.456 -.539 .0646 -.014 .091 .217 .283 -.253 -.332 -.363 -.430 -.466 .0065 -.101 -.018 .092 .223 .283 -.253 -.332 -.359 -.436 -.0334 -.152 -.125 -.014 .092 .217 .272 -.260 -.293 -.369 -.0133 -.133 -.155 -.125 -.018 .091 .214 .267 -.267 -.305 .0162 -.115 -.133 -.152 -.101 -.014 .082 .212 .263 -.244 .039


Ace -.044 -.049 -.072 -.084 .011 .126 .236 .378 .486 -.02110 -.488 -.609 -.738 -.873 -.931 -tOn -1.265 -1.499 -1.693 .4869 .524 -.534 -.587 -.718 -.861 -.912 -1.060 -1.255 -1.499 .3788 .423 .532 -.521 -.663 -.726 -.781 -.908 -1.060 -1.265 .2367 .116 .425 .541 -.506 -.665 -.813 -.781 -.912 -1.077 .1266 -.165 .110 .431 .566 -.506 -.665 -.726 -.861 -.931 .0115 -.404 -.170 .112 .445 .566 -.506 -.663 -.718 -.873 -.0844 -.563 -.431 -.186 .112 .431 .541 -.521 -.587 -.738 -.0723 -.589 -.570 -.431 -.170 .110 .425 .532 -.534 -.609 -.0492 -.584 -.589 -.563 -.404 -.165 .116 .423 .524 -.488 -.044

EV Split2 3 4 5 6 7 8 9 10 Ace

reg -.130 -.204 -.238 -.234 -.222 -.165 .015 .168 .309 .565dbs -.041 -.120 -.152 -.160 -.129 -.075 .087 .201 .309 .565

ApPENDIX E 305

1 Deck, H17, Dealer Shows 2EV Stand

2 3 4 5 6 7 8 9 10 Ace

Ace -.279 -.280 -.280 -.262 -.135 .127 .395 .652 .878 -.27210 -.305 -.306 -.306 -.289 -.292 -.162 .107 .376 .622 .8789 -.281 -.282 -.282 -.264 -.268 -.271 -.141 .125 .376 .6528 -.276 -.278 -.277 -.259 -.262 -.267 -.270 -.141 .107 .3957 -.274 -.275 -.274 -.256 -.260 -.263 -.267 -.271 -.162 .1276 -.271 -.273 -.271 -.253 -.256 -.260 -.262 -.268 -.292 -.135S -.268 -.270 -.269 -.250 -.253 -.256 -.259 -.264 -.289 -.2624 -.286 -.288 -.287 -.269 -.271 -.274 -.2n -.282 -.306 -.2803 -.287 -.289 -.288 -.270 -.273 -.275 -.278 -.282 -.306 -.2802 -.285 -.287 -.286 -.268 -.271 -.274 -.276 -.281 -.305 -.279

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .040 .018 -.011 -.031 .006 .062 .117 .189 .246 .09310 -.244 -.305 -.370 -.438 -.467 -.540 -.634 -.750 -.847 .2469 .262 -.267 -.294 -.360 -.432 -.458 -.532 -.628 -.750 .1898 .211 .267 -.261 -.332 -.364 -.392 -.456 -.532 -.634 .1177 .080 .212 .271 -.253 -.333 -.408 -.392 -.458 -.540 .0626 -.017 .089 .216 .282 -.253 -.333 -.364 -.432 -.467 .0065 -.100 -.020 .091 .221 .282 -.253 -,332 -.360 -.438 -.0314 -.149 -.124 -.015 .091 .216 .271 -.261 -.294 -.370 -.0113 -.129 -.150 -.124 -.020 .089 .212 .267 -.267 -.305 .0182 -.112 -.129 -.149 -.100 -.017 .080 .211 .262 -.244 .040


Ace -.041 -.045 -.068 -.079 .012 .123 .233 .377 .485 -.01910 -.489 -.611 -.740 -.876 -.935 -1.080 -1.267 -1.500 -1.694 .4859 .523 -.534 -.588 -.721 -.864 -.916 -1.064 -1.257 -1.500 .3n8 .422 .532 -.521 -.665 -.728 -.785 -.912 -1.064 -1.267 .2337 .114 .423 .541 -.507 -.666 -.816 -.785 -.916 -1.080 .1236 -.166 .108 .430 .565 -.506 -.666 -.728 -.864 -.935 .0125 -.398 -.171 .111 .443 .565 -.507 -.665 -.721 -.876 -.0794 -.552 -.426 -.187 .111 .430 .541 -.521 -.588 -.740 -.0683 -.574 -.556 -.426 -.171 .108 .423 .532 -.534 -.611 -.0452 -.569 -.574 -.552 -.398 -.166 .114 .422 .523 -.489 -.041

EV Split2 3 4 5 6 7 8 9 10 Ace

reg -.130 -.199 -.234 -.228 -.214 -.165 .009 .164 .306 .564dbs -.041 -.114 -.148 -.153 -.122 -.076 .081 .197 .306 .564



2 3 4 S 6 7 8 9 10 Ace

Ace -.242 -.241 -.225 -.222 -.094 .166 .420 .644 .884 -.23310 -.269 -.267 -.252 -.249 -.251 -.120 .144 .384 .636 .8849 -.265 -.263 -.247 -.245 -.247 -.251 -.122 .122 .384 .6448 -.239 -.237 -.221 -.218 -.221 -.225 -.230 -.122 .144 .4207 -.235 -.233 -.217 -.214 -.216 -.221 -.225 -.251 -.120 .1666 -.232 -.229 -.213 -.210 -.212 -.216 -.221 -.247 -.251 -.0945 -.230 -.228 -.211 -.207 -.210 -.214 -.218 -.245 -.249 -.2224 -.233 -.231 -.215 -.211 -.213 -.217 -.221 -.247 -.252 -.2253 -.249 -.247 -.231 -.228 -.229 -.233 -.237 -.263 -.267 -.2412 -.252 -.249 -.233 -.230 -.232 -.235 -.239 -.265 -.269 -.242

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .071 .044 .022 -.004 .035 .093 .172 .196 .270 .12010 -.220 -.284 -.356 -.430 -.461 -.536 -.634 -.713 -.846 .2709 .269 -.256 -.281 -.356 -.432 -.459 -.536 -.638 -.713 .1968 .241 .294 -.230 -.306 -.347 -.381 -.450 -.536 -.634 .1727 .130 .246 .304 -.222 -.304 -.389 -.381 -.459 -.536 .0936 .022 .138 .249 .313 -.223 -.304 -.347 -.432 -.461 .0355 -.063 .018 .139 .253 .313 -.222 -.306 -.356 -.430 -.0044 -.108 -.084 .027 .139 .249 .304 -.230 -.281 -.356 .0223 -.100 -.120 -.084 .018 .138 .246 .294 -.256 -.284 .0442 -.083 -.100 -.108 -.063 .022 .130 .241 .269 -.220 .071

EV Double Down2 3 4 S 6 7 8 9 10 Ace

Ace .027 .009 .001 -.021 .072 .188 .345 .391 .535 .05310 -.439 -.567 -.712 -.860 -.923 -1.073 -1.267 -1.425 -1.693 .5359 .537 -.512 -.561 -.713 -.865 -.918 -1.071 -1.277 -1.425 .3918 .482 .589 -.460 -.612 -.694 -.763 -.900 -1.071 -1.267 .3457 .193 .491 .610 -.445 -.609 -.n7 -.763 -.918 -1.073 .1886 -.080 .188 .500 .628 -.445 -.609 -.694 -.865 -.923 .0725 -.313 -.085 .194 .509 .628 -.445 -.612 -.713 -.860 -.0214 -.448 -.326 -.084 .194 .500 .610 -.460 -.561 -.712 .0013 -.499 -.474 -.326 -.085 .188 .491 .589 -.512 -.567 .0092 -.503 -.499 -.448 -.313 -.080 .193 .482 .537 -.439 .027

EV Split2 3 4 5 6 7 8 9 10 Ace

reg -.072 -.129 -.129 -.153 -.137 -.086 .075 .169 .363 .612dbs .015 -.038 -.033 -.075 -.038 .000 .155 .210 .363 .612

ApPENDIX E 307


2 3 4 5 6 7 8 9 10 Ace

Ace -.238 -.238 -.221 -.218 -.095 .160 .415 .641 .883 -.23110 -.264 -.263 -.246 -.243 -.246 -.122 .135 .377 .632 .8839 -.260 -.259 -.242 -.240 -.242 -.246 -.124 .114 .377 .6418 -.234 -.233 -.216 -.213 -.216 -.220 -.225 -.124 .135 .4157 -.230 -.229 -.211 -.208 -.211 -.216 -.220 -.246 -.122 .1606 -.227 -.225 -.208 -.204 -.207 -.211 -.216 -.242 -.246 -.095S -.225 -.224 -.205 -.202 -.204 -.208 -.213 -.240 -.243 -.2184 -.228 -.227 -.209 -.205 -.208 -.211 -.216 -.242 -.246 -.2213 -.246 -.245 -.227 -.224 -.225 -.229 -.233 -.259 -.263 -.2382 -.246 -.246 -.228 -.225 -.227 -.230 -.234 -.260 -.264 -.238

EV Hit2 3 4 S 6 7 8 9 10 Ace

Ace .071 .044 .023 -.003 .035 .091 .171 .195 .270 .11910 -.220 -.284 -.357 -.431 -.463 -.538 -.634 -.713 -.846 .2709 .268 -.256 -.281 -.357 -.434 -.461 -.537 -.639 -.713 .1958 .240 .294 -.230 -.306 -.348 -.383 -.451 -.537 -.634 .1717 .129 .245 .304 -.223 -.305 -.389 -.383 -.461 -.538 .0916 .020 .137 .248 .312 -.223 -.305 -.348 -.434 -.463 .035S -.063 .016 .138 .252 .312 -.223 -.306 -.357 -.431 -.0034 -.105 -.083 .026 .138 .248 .304 -.230 -.281 -.357 .0233 -.098 -.118 -.083 .016 .137 .245 .294 -.256 -.284 .0442 -.081 -.098 -.105 -.063 .020 .129 .240 .268 -.220 .071


Ace .029 .011 .005 -.018 .073 .185 .343 .390 .535 .05510 -.440 -.568 -.713 -.862 -.926 -1.075 -1.269 -1.426 -1.693 .5359 .537 -.513 -.563 -.714 -.867 -.921 -1.074 -1.278 -1.426 .3908 .480 .589 -.461 -.613 -.696 -.765 -.903 -1.074 -1.269 .3437 .192 .490 .610 -.445 -.610 -.779 -.765 -.921 -1.075 .1856 -.081 .187 .499 .628 -.445 -.610 -.696 -.867 -.926 .073S -.308 -.085 .193 .507 .628 -.445 -.613 -.714 -.862 -.0184 -.437 -.322 -.084 .193 .499 .610 -.461 -.563 -.713 .0053 -.491 -.470 -.322 -.085 .187 .490 .589 -.513 -.568 .0112 -.493 -.491 -.437 -.308 -.081 .192 .480 .537 -.440 .029

EV Split2 3 4 S 6 7 8 9 10 Ace

reg -.071 -.128 -.124 -.148 -.132 -.086 .071 .166 .361 .612dbs .015 -.036 -.027 -.070 -.032 .000 .151 .208 .361 .612



2 3 4 5 6 7 8 9 10 Ace

Ace -.187 -.171 -.167 -.164 -.037 .204 .415 .654 .884 -.17910 -.213 -.198 -.194 -.192 -.194 -.065 .164 .404 .645 .8849 -.208 -.192 -.189 -.187 -.190 -.195 -.085 .167 .404 .6548 -.203 -.188 -.184 -.182 -.185 -.191 -.216 -.085 .164 .4157 -.178 -.162 -.158 -.156 -.159 -.165 -.191 -.195 -.065 .2046 -.172 -.156 -.153 -.150 -.153 -.159 -.185 -.190 -.194 -.0375 -.170 -.154 -.150 -.147 -.150 -.156 -.182 -.187 -.192 -.1644 -.173 -.157 -.153 -.150 -.153 -.158 -.184 -.189 -.194 -.1673 -.177 -.160 -.157 -.154 -.156 -.162 -.188 -.192 -.198 -.1712 -.193 -.177 -.173 -.170 -.172 -.178 -.203 -.208 -.213 -.187

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .110 .091 .061 .037 .077 .156 .186 .229 .297 .14210 -.194 -.258 -.337 -.418 -.455 -.535 -.597 -.712 -.846 .2979 .300 -.226 -.255 -.336 -.420 -.453 -.541 -.597 -.712 .2298 .252 .312 -.214 -.292 -.338 -.380 -.461 -.541 -.597 .1867 .167 .285 .342 -.192 -.277 -.369 -.380 -.453 -.535 .1566 .080 .185 .292 .352 -.190 -.277 -.338 -.420 -.455 .077S -.011 .086 .189 .295 .352 -.192 -.292 -.336 -.418 .0374 -.056 -.018 .097 .189 .292 .342 -.214 -.255 -.337 .0613 -.042 -.048 -.018 .086 .185 .285 .312 -.226 -.258 .0912 -.036 -.042 -.056 -.011 .080 .167 .252 .300 -.194 .110


Ace .114 .108 .084 .061 .153 .312 .372 .458 .594 .13510 -.389 -.516 -.673 -.836 -.909 -1.070 -1.195 -1.423 -1.693 .5949 .599 -.452 -.509 -.671 -.840 -.906 -1.082 -1.194 -1.423 .4588 .504 .624 -.429 -.584 -.676 -.760 -.922 -1.082 -1.195 .3727 .271 .570 .684 -.384 -.555 -.738 -.760 -.906 -1.070 .3126 .022 .289 .585 .703 -.381 -.555 -.676 -.840 -.909 .1535 -.196 .040 .296 .589 .703 -.384 -.584 -.671 -.836 .0614 -.325 -.179 .043 .296 .585 .684 -.429 -.509 -.673 .0843 -.353 -.304 -.179 .040 .289 .570 .624 -.452 -.516 .1082 -.386 -.353 -.325 -.196 .022 .271 .504 .599 -.389 .114

EV Split2 3 4 5 6 7 8 9 10 Ace

reg -.009 .008 -.015 -.040 -.029 .015 .104 .252 .416 .668dbs .105 .134 .121 .068 .110 .153 .194 .298 .416 .668

ApPENDIX E 309


2 3 4 5 6 7 8 9 10 Ace

Ace -.183 -.166 -.162 -.160 -.039 .196 .410 .651 .883 -.17610 -.208 -.191 -.188 -.185 -.188 -.068 .154 .396 .640 .8839 -.203 -.186 -.183 -.181 -.184 -.189 -.088 .156 .396 .6518 -.199 -.181 -.177 -.176 -.179 -.185 -.210 -.088 .154 .4107 -.173 -.155 -.152 -.149 -.153 -.159 -.185 -.189 -.068 .1966 -.167 -.150 -.146 -.144 -.147 -.153 -.179 -.184 -.188 -.0395 -.165 -.147 -.143 -.141 -.144 -.149 -.176 -.181 -.185 -.1604 -.168 -.150 -.146 -.143 -.146 -.152 -.177 -.183 -.188 -.1623 -.171 -.154 -.150 -.147 -.150 -.155 -.181 -.186 -.191 -.1662 -.189 -.171 -.168 -.165 -.167 -.173 -.199 -.203 -.208 -.183

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .111 .092 .063 .040 .077 .154 .185 .229 .297 .14310 -.195 -.259 -.337 -.420 -.456 -.537 -.598 -.712 -.846 .2979 .299 -.226 -.255 -.337 -.421 -.455 -.543 -.598 -.712 .2298 .252 .312 -.215 -.293 -.339 -.382 -.463 -.543 -.598 .1857 .166 .284 .342 -.192 -.278 -.370 -.382 -.455 -.537 .1546 .079 .184 .292 .351 -.190 -.278 -.339 -.421 -.456 .077S -.010 .084 .188 .294 .351 -.192 -.293 -.337 -.420 .0404 -.053 -.016 .096 .188 .292 .342 -.215 -.255 -.337 .0633 -.039 -.044 -.016 .084 .184 .284 .312 -.226 -.259 .0922 -.034 -.039 -.053 -.010 .079 .166 .252 .299 -.195 .111


Ace .116 .112 .088 .065 .154 .309 .370 .457 .594 .13710 -.389 -.518 -.675 -.839 -.913 -1.073 -1.196 -1.424 -1.693 .5949 .599 -.452 -.510 -.673 -.842 -.910 -1.085 -1.196 -1.424 .4578 .503 .624 -.429 -.585 -.678 -.763 -.926 -1.085 -1.196 .3707 .269 .568 .685 -.385 -.556 -.740 -.763 -.910 -1.073 .3096 .021 .287 .583 .703 -.381 -.556 -.678 -.842 -.913 .1545 -.192 .039 .295 .587 .703 -.385 -.585 -.673 -.839 .0654 -.316 -.173 .043 .295 .583 .685 -.429 -.510 -.675 .0883 -.343 -.291 -.173 .039 .287 .568 .624 -.452 -.518 .1122 -.379 -.343 -.316 -.192 .021 .269 .503 .599 -.389 .116

EV Split2 3 4 5 6 7 8 9 10 Ace

reg -.006 .015 -.007 -.032 -.022 .016 .100 .249 .415 .668dbs .108 .141 .129 .076 .117 .155 .190 .296 .415 .668



2 3 4 5 6 7 8 9 10 Ace

Ace -.119 -.117 -.114 -.111 .004 .222 .461 .682 .893 -.13110 -.145 -.142 -.139 -.136 -.142 -.044 .202 .448 .674 .8939 -.139 -.138 -.135 -.132 -.138 -.164 -.045 .203 .448 .6828 -.136 -.133 -.132 -.129 -.135 -.161 -.166 -.045 .202 .4617 -.131 -.128 -.125 -.124 -.130 -.156 -.161 -.164 -.044 .2226 -.104 -.102 -.099 -.096 -.103 -.130 -.135 -.138 -.142 .0045 -.099 -.096 -.093 -.091 -.096 -.124 -.129 -.132 -.136 -.1114 -.102 -.099 -.096 -.093 -.099 -.125 -.132 -.135 -.139 -.1143 -.105 -.102 -.099 -.096 -.102 -.128 -.133 -.138 -.142 -.1172 -.108 -.105 -.102 -.099 -.104 -.131 -.136 -.139 -.145 -.119

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .159 .136 .108 .082 .140 .174 .226 .268 .331 .18210 -.164 -.236 -.315 -.400 -.444 -.492 -.591 -.709 -.846 .3319 .340 -.197 -.232 -.314 -.402 -.452 -.493 -.590 -.709 .2688 .296 .349 -.187 -.270 -.317 -.373 -.453 -.493 -.591 .2267 .195 .307 .361 -.178 -.269 -.371 -.373 -.452 -.492 .1746 .130 .234 .339 .393 -.163 -.269 -.317 -.402 -.444 .1405 .056 .140 .246 .347 .393 -.178 -.270 -.314 -.400 .0824 .008 .048 .153 .246 .339 .361 -.187 -.232 -.315 .1083 .021 .008 .048 .140 .234 .307 .349 -.197 -.236 .1362 .035 .021 .008 .056 .130 .195 .296 .340 -.164 .159


Ace .211 .203 .174 .147 .279 .348 .452 .535 .662 .21510 -.328 -.471 -.630 -.799 -.889 -.985 -1.182 -1.417 -1.691 .6629 .679 -.394 -.465 -.628 -.804 -.904 -.985 -1.181 -1.417 .5358 .591 .698 -.374 -.541 -.635 -.747 -.906 -.985 -1.182 .4527 .331 .614 .723 -.356 -.539 -.741 -.747 -.904 -.985 .3486 .130 .391 .677 .787 -.325 -.539 -.635 -.804 -.889 .2795 -.072 .154 .414 .694 .787 -.356 -.541 -.628 -.799 .1474 -.187 -.063 .161 .414 .677 .723 -.374 -.465 -.630 .1743 -.210 -.185 -.063 .154 .391 .614 .698 -.394 -.471 .2032 -.215 -.210 -.187 -.072 .130 .331 .591 .679 -.328 .211

EV Split2 3 4 5 6 7 8 9 10 Ace

reg .127 .116 .094 .067 .066 .038 .187 .338 .460 .732dbs .272 .270 .252 .196 .236 .192 .290 .391 .460 .732

ApPENDIX E 311


2 3 4 5 6 7 8 9 10 Ace

Ace -.118 -.115 -.113 -.110 .004 .220 .459 .681 .892 -.13010 -.142 -.140 -.137 -.134 -.140 -.045 .198 .445 .672 .8929 -.137 -.135 -.132 -.130 -.136 -.162 -.046 .199 .445 .6818 -.134 -.131 -.129 -.127 -.132 -.159 -.164 -.046 .198 .4597 -.129 -.126 -.123 -.122 -.127 -.154 -.159 -.162 -.045 .2206 -.102 -.099 -.096 -.094 -.101 -.127 -.132 -.136 -.140 .004S -.097 -.094 -.091 -.089 -.094 -.122 -.127 -.130 -.134 -.1104 -.100 -.096 -.093 -.091 -.096 -.123 -.129 -.132 -.137 -.1133 -.102 -.100 -.096 -.094 -.099 -.126 -.131 -.135 -.140 -.1152 -.105 -.102 -.100 -.097 -.102 -.129 -.134 -.137 -.142 -.118

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .159 .137 .108 .082 .140 .174 .226 .268 .331 .18210 -.164 -.236 -.315 -.400 -.445 -.493 -.591 -.709 -.846 .3319 .339 -.197 -.232 -.314 -.403 -.453 -.493 -.591 -.709 .2688 .295 .349 -.187 -.270 -.318 -.374 -.454 -.493 -.591 .2267 .194 .307 .361 -.178 -.269 -.371 -.374 -.453 -.493 .1746 .129 .234 .338 .393 -.163 -.269 -.318 -.403 -.445 .1405 .056 .140 .245 .347 .393 -.178 -.270 -.314 -.400 .0824 .010 .049 .153 .245 .338 .361 -.187 -.232 -.315 .1083 .022 .009 .049 .140 .234 .307 .349 -.197 -.236 .1372 .036 .022 .010 .056 .129 .194 .295 .339 -.164 .159


Ace .212 .203 .175 .148 .279 .347 .452 .535 .662 .21510 -.328 -.472 -.630 -.801 -.890 -.986 -1.182 -1.418 -1.691 .6629 .679 -.394 -.465 -.629 -.806 -.906 -.986 -1.181 -1.418 .5358 .591 .698 -.374 -.541 -.636 -.748 -.907 -.986 -1.182 .4527 .330 .614 .723 -.356 -.539 -.742 -.748 -.906 -.986 .3476 .129 .390 .676 .786 -.325 -.539 -.636 -.806 -.890 .279.5 -.070 .154 .414 .693 .786 -.356 -.541 -.629 -.801 .1484 -.182 -.060 .161 .414 .676 .723 -.374 -.465 -.630 .1753 -.205 -.181 -.060 .154 .390 .614 .698 -.394 -.472 .2032 -.210 -.205 -.182 -.070 .129 .330 .591 .679 -.328 .212

EV Split2 3 4 5 6 7 8 9 10 Ace

reg .129 .118 .096 .069 .069 .038 .185 .337 .495 .732dbs .274 .272 .255 .199 .238 .192 .288 .390 .495 .732



2 3 4 5 6 7 8 9 10 Ace

Ace -.115 -.112 -.110 -.108 .010 .262 .482 .694 .899 -.10410 -.161 -.158 -.156 -.155 -.180 -.012 .268 .484 .697 .8999 -.157 -.154 -.152 -.151 -.1n -.180 -.012 .265 .484 .6948 -.152 -.150 -.148 -.147 -.173 -.178 -.179 -.012 .268 .4827 -.149 -.147 -.145 -.144 -.170 -.175 -.178 -.180 -.012 .2626 -.145 -.142 -.140 -.140 -.166 -.170 -.173 -.1n -.180 .010S -.118 -.116 -.113 -.112 -.140 -.144 -.147 -.151 -.155 -.1084 -.120 -.117 -.115 -.113 -.140 -.145 -.148 -.152 -.156 -.1103 -.122 -.119 -.117 -.116 -.142 -.147 -.150 -.154 -.158 -.1122 -.125 -.122 -.120 -.118 -.145 -.149 -.152 -.157 -.161 -.115

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .169 .147 .120 .116 .133 .192 .241 .280 .341 .20010 -.160 -.229 -.308 -.387 -.396 -.483 -.586 -.707 -.845 .3419 .346 -.194 -.228 -.309 -.401 -.397 -.484 -.587 -.707 .2808 .307 .356 -.184 -.268 -.323 -.362 -.397 -.484 -.586 .2417 .209 .318 .366 -.179 -.281 -.367 -.362 -.397 -.483 .1926 .132 .231 .331 .380 -.194 -.281 -.323 -.401 -.396 .133S .069 .163 .263 .362 .380 -.179 -.268 -.309 -.387 .1164 .013 .059 .175 .263 .331 .366 -.184 -.228 -.308 .1203 .019 .013 .059 .163 .231 .318 .356 -.194 -.229 .1472 .031 .019 .013 .069 .132 .209 .307 .346 -.160 .169


Ace .230 .221 .200 .216 .266 .384 .482 .559 .682 .24710 -.319 -.458 -.616 -.n5 -.793 -.967 -1.172 -1.414 -1.690 .6829 .691 -.388 -.457 -.618 -.802 -.795 -.968 -1.173 -1.414 .5598 .615 .712 -.369 -.536 -.647 -.724 -.793 -.968 -1.172 .4827 .357 .636 .731 -.358 -.562 -.734 -.724 -.795 -.967 .3846 .124 .379 .663 .761 -.388 -.562 -.647 -.802 -.793 .266S -.062 .189 .443 .723 .761 -.358 -.536 -.618 -.n5 .2164 -.216 -.058 .193 .443 .663 .731 -.369 -.457 -.616 .2003 -.245 -.215 -.058 .189 .379 .636 .712 -.388 -.458 .2212 -.250 -.245 -.216 -.062 .124 .357 .615 .691 -.319 .230

EV Split2 3 4 S 6 7 8 9 10 Ace

reg .125 .111 .083 .055 -.012 .052 .233 .359 .520 .758dbs .278 .263 .240 .190 .154 .209 .339 .413 .520 .758

ApPENDIX E 313


2 J 4 5 6 7 8 9 10 Ace

Ace -.087 -.084 -.082 -.081 -.002 .215 .447 .673 .892 -.08510 -.124 -.122 -.119 -.120 -.145 -.031 .203 .436 .668 .8929 -.122 -.120 -.117 -.117 -.144 -.148 -.032 .201 .436 .6738 -.117 -.115 -.113 -.113 -.140 -.145 -.146 -.032 .203 .4477 -.114 -.111 -.110 -.110 -.137 -.142 -.145 -.148 -.031 .2156 -.109 -.106 -.104 -.105 -.132 -.137 -.140 -.144 -.145 -.002S -.082 -.079 -.077 -.077 -.105 -.110 -.113 -.117 -.120 -.0814 -.082 -.079 -.076 -.on -.104 -.110 -.113 -.117 -.119 -.0823 -.085 -.082 -.079 -.079 -.106 -.111 -.115 -.120 -.122 -.0842 -.087 -.085 -.082 -.082 -.109 -.114 -.117 -.122 -.124 -.087

EV Hit2 3 4 S 6 7 8 9 10 Ace

Ace .174 .154 .129 .124 .131 .181 .233 .275 .339 .20210 -.161 -.233 -.314 -.398 -.409 -.494 -.592 -.710 -.846 .3399 .344 -.195 -.231 -.316 -.412 -.410 -.494 -.593 -.710 .2758 .301 .355 -.185 -.271 -.331 -.373 -.409 -.494 -.592 .2337 .199 .312 .366 -.179 -.284 -.375 -.373 -.410 -.494 .1816 .119 .221 .326 .379 -.192 -.284 -.331 -.412 -.409 .131S .072 .151 .253 .354 .379 -.179 -.271 -.316 -.398 .1244 .036 .064 .164 .253 .326 .366 -.185 -.231 -.314 .1293 .040 .036 .064 .151 .221 .312 .355 -.195 -.233 .1542 .050 .040 .036 .072 .119 .199 .301 .344 -.161 .174


Ace .245 .239 .218 .232 .262 .361 .466 .550 .679 .25810 -.322 -.467 -.627 -.795 -.818 -.988 -1.185 -1.419 -1.691 .6799 .688 -.390 -.463 -.631 -.823 -.820 -.989 -1.186 -1.419 .5508 .603 .710 -.369 -.542 -.661 -.746 -.819 -.989 -1.185 .4667 .340 .624 .732 -.358 -.568 -.749 -.746 -.820 -.988 .3616 .110 .364 .651 .757 -.385 -.568 -.661 -.823 -.818 .2625 -.037 .178 .430 .708 .757 -.358 -.542 -.631 -.795 .2324 -.147 -.029 .185 .430 .651 .732 -.369 -.463 -.627 .2183 -.169 -.146 -.029 .178 .364 .624 .710 -.390 -.467 .2392 -.175 -.169 -.147 -.037 .110 .340 .603 .688 -.322 .245

EV Split2 3 4 S 6 7 8 9 10 Ace

reg .157 .147 .125 .092 .022 .051 .202 .339 .507 .756dbs .309 .300 .283 .227 .186 .209 .307 .393 .507 .756


1 Deck, Dealer Shows 7EV Stand

2 3 4 5 6 7 8 9 10 Ace

Ace -.462 -.461 -.458 -.468 -.090 .412 .614 .773 .925 -.453to -.473 -.471 -.469 -.479 -.484 -.122 .389 .610 .765 .9259 -.494 -.492 -.490 -.500 -.503 -.508 -.123 .401 .610 .7738 -.488 -.486 -.484 -.494 -.499 -.502 -.503 -.123 .389 .6147 -.486 -.485 -.483 -.493 -.498 -.502 -.502 -.508 -.122 .4126 -.482 -.480 -.479 -.489 -.494 -.498 -.499 -.503 -.484 -.0905 -.477 -.476 -.473 -.485 -.489 -.493 -.494 -.500 -.479 -.4684 -.467 -.465 -.463 -.473 -.479 -.483 -.484 -.490 -.469 -.4583 -.469 -.468 -.465 -.476 -.480 -.485 -.486 -.492 -.471 -.4612 -.471 -.469 -.467 -.477 -.482 -.486 -.488 -.494 -.473 -.462

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .108 .061 .034 -.023 .060 .175 .222 .243 .286 .15910 -.212 -.270 -.342 -.365 -.376 -.452 -.567 -.699 -.843 .2869 .289 -.247 -.274 -.348 -.363 -.375 -.449 -.566 -.699 .2438 .268 .292 -.245 -.327 -.349 -.324 -.374 -.449 -.567 .2227 .184 .278 .294 -.258 -.331 -.389 -.324 -.375 -.452 .1756 .093 .198 .286 .298 -.265 -.331 -.349 -.363 -.376 .0605 -.066 .093 .202 .279 .298 -.258 -.327 -.348 -.365 -.0234 -.162 -.069 .112 .202 .286 .294 -.245 -.274 -.342 .0343 -.118 -.163 -.069 .093 .198 .278 .292 -.247 -.270 .0612 -.090 -.118 -.162 -.066 .093 .184 .268 .289 -.212 .108


Ace -.158 -.175 -.141 -.190 .014 .240 .325 .351 .467 -.13710 -.496 -.583 -.707 -.736 -.752 -.904 -1.135 -1.397 -1.686 .4679 .455 -.579 -.600 -.728 -.745 -.750 -.898 -1.132 -1.397 .3518 .416 .472 -.564 -.707 -.735 -.666 -.747 -.898 -1.135 .3257 .160 .444 .487 -.585 -.714 -.823 -.666 -.750 -.904 .2406 -.149 .175 .475 .500 -.599 -.714 -.735 -.745 -.752 .0145 -.568 -.133 .190 .466 .500 -.585 -.707 -.728 -.736 -.1904 -.871 -.543 -.109 .190 .475 .487 -.564 -.600 -.707 -.1413 -.938 -.872 -.543 -.133 .175 .444 .472 -.579 -.583 -.1752 -.943 -.938 -.871 -.568 -.149 .160 .416 .455 -.496 -.158

EV Split2 3 4 5 6 7 8 9 10 Ace

reg -.053 -.114 -.222 -.298 -.269 -.122 .202 .341 .482 .540dbs .006 -.056 -.158 -.244 -.201 -.066 .250 .363 .482 .540

ApPENDIX E 315


2 3 4 5 6 7 8 9 10 Ace

Ace -.508 -.505 -.515 -.515 -.385 .121 .608 .785 .928 -.50010 -.518 -.515 -.524 -.525 -.527 -.394 .095 .577 .783 .9289 -.514 -.512 -.521 -.521 -.523 -.526 -.415 .064 .577 .7858 -.534 -.532 -.541 -.542 -.543 -.546 -.552 -.415 .095 .6087 -.529 -.526 -.536 -.536 -.539 -.540 -.546 -.526 -.394 .1216 -.525 -.524 -.533 -.534 -.536 -.539 -.543 -.523 -.527 -.3855 -.523 -.520 -.531 -.531 -.534 -.536 -.542 -.521 -.525 -.5154 -.523 -.520 -.529 -.531 -.533 -.536 -.541 -.521 -.524 -.5153 -.513 -.511 -.520 -.520 -.524 -.526 -.532 -.512 -.515 -.5052 -.515 -.513 -.522 -.523 -.525 -.529 -.534 -.514 -.518 -.508

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .040 .035 -.035 -.084 -.065 .048 .158 .172 .221 .09410 -.274 -.328 -.357 -.418 -.425 -.474 -.565 -.697 -.843 .2219 .216 -.316 -.339 -.370 -.421 -.428 -.475 -.566 -.697 .1728 .207 .221 -.319 -.387 -.369 -.380 -.426 -.475 -.565 .1587 .108 .217 .222 -.321 -.394 -.408 -.380 -.428 -.474 .0486 -.055 .118 .208 .230 -.322 -.394 -.369 -.421 -.425 -.0655 -.217 -.056 .109 .208 .230 -.321 -.387 -.370 -.418 -.0844 -.233 -.227 -.054 .109 .208 .222 -.319 -.339 -.357 -.0353 -.180 -.230 -.227 -.056 .118 .217 .221 -.316 -.328 .0352 -.140 -.180 -.233 -.217 -.055 .108 .207 .216 -.274 .040


Ace -.313 -.254 -.314 -.333 -.230 -.016 .190 .230 .332 -.29610 -.626 -.706 -.739 -.844 -.850 -.948 -1.130 -1.395 -1.685 .3329 .328 -.704 -.724 -.762 -.856 -.856 -.951 -1.133 -1.395 .2308 .294 .330 -.719 -.829 -.775 -.777 -.853 -.951 -1.130 .1907 .007 .327 .340 -.711 -.843 -.858 -.n7 -.856 -.948 -.0166 -.437 .014 .317 .365 -.712 -.843 -.775 -.856 -.850 -.2305 -.854 -.428 .000 .323 .365 -.711 -.829 -.762 -.844 -.3334 -1.024 -.849 -.447 .000 .317 .340 -.719 -.724 -.739 -.3143 -1.026 -1.000 -.849 -.428 .014 .327 .330 -.704 -.706 -.2542 -1.030 -1.026 -1.024 -.854 -.437 .007 .294 .328 -.626 -.313

EV Split2 3 4 5 6 7 8 9 10 Ace

reg -.212 -.265 -.342 -.448 -.412 -.423 -.100 .179 .354 .406dbs -.176 -.225 -.299 -.412 -.368 -.385 -.074 .195 .354 .406



2 3 4 5 6 7 8 9 10 Ace

Ace -.517 -.527 -.526 -.526 -.407 -.179 .288 .766 .936 -.51110 -.528 -.536 -.536 -.535 -.539 -.416 -.196 .264 .744 .9369 -.524 -.534 -.532 -.532 -.536 -.540 -.412 -.196 .264 .7668 -.520 -.530 -.529 -.528 -.532 -.536 -.517 -.412 -.196 .2887 -.541 -.550 -.550 -.549 -.551 -.556 -.536 -.540 -.416 -.1796 -.537 -.546 -.546 -.545 -.549 -.551 -.532 -.536 -.539 -.407S -.532 -.542 -.542 -.541 -.545 -.549 -.528 -.532 -.535 -.5264 -.533 -.542 -.543 -.542 -.546 -.550 -.529 -.532 -.536 -.5263 -.533 -.543 -.542 -.542 -.546 -.550 -.530 -.534 -.536 -.5272 -.524 -.533 -.533 -.532 -.537 -.541 -.520 -.524 -.528 -.517

EV Hit2 3 4 S 6 7 8 9 10 Ace

Ace -.013 -.059 -.113 -.166 -.134 -.087 .005 .097 .149 -.00110 -.344 -.359 -.414 -.475 -.479 -.526 -.593 -.698 -.842 .1499 .143 -.392 -.365 -.426 -.480 -.482 -.532 -.595 -.698 .0978 .121 .139 -.395 -.413 -.437 -.443 -.487 -.532 -.593 .0057 -.052 .118 .150 -.390 -.419 -.475 -.443 -.482 -.526 -.0876 -.208 -.051 .118 .152 -.386 -.419 -.437 -.480 -.479 -.134S -.284 -.218 -.051 .119 .152 -.390 -.413 -.426 -.475 -.1664 -.303 -.304 -.206 -.051 .118 .150 -.395 -.365 -.414 -.1133 -.261 -.309 -.304 -.218 -.051 .118 .139 -.392 -.359 -.0592 -.221 -.261 -.303 -.284 -.208 -.052 .121 .143 -.344 -.013


Ace -.373 -.394 -.422 -.452 -.345 -.255 -.060 .110 .216 -.42110 -.746 -.752 -.839 -.952 -.959 -1.053 -1.186 -1.396 -1.684 .2169 .214 -.840 -.759 -.860 -.967 -.964 -1.064 -1.189 -1.396 .1108 .174 .215 -.834 -.848 -.888 -.894 -.974 -1.064 -1.186 -.0607 -.274 .154 .225 -.826 -.869 -.978 -.894 -.964 -1.053 -.2556 -.692 -.290 .165 .240 -.817 -.869 -.888 -.967 -.959 -.345S -.927 -.703 -.279 .174 .240 -.826 -.848 -.860 -.952 -.4524 -1.046 -.946 -.701 -.279 .165 .225 -.834 -.759 -.839 -.4223 -1.066 -1.066 -.946 -.703 -.290 .154 .215 -.840 -.752 -.3942 -1.047 -1.066 -1.046 -.927 -.692 -.274 .174 .214 -.746 -.373

EV Split2 3 4 5 6 7 8 9 10 Ace

reg -.382 -.417 -.492 -.607 -.569 -.564 -.430 -.112 .184 .290dbs -.358 -.396 -.468 -.584 -.544 -.545 -.406 -.103 .184 .290

ApPENDIX E 317


2 3 4 5 6 7 8 9 10 Ace

Ace -.539 -.538 -.538 -.537 -.418 -.186 .064 .555 .958 -.53110 -.540 -.539 -.539 -.539 -.543 -.412 -.155 .103 .583 .9589 -.537 -.535 -.536 -.535 -.539 -.518 -.391 -.133 .103 .5558 -.533 -.532 -.532 -.531 -.535 -.514 -.518 -.391 -.155 .0647 -.529 -.528 -.528 -.527 -.531 -.510 -.514 -.518 -.412 -.1866 -.551 -.550 -.551 -.550 -.552 -.531 -.535 -.539 -.543 -.418S -.547 -.546 -.546 -.545 -.550 -.527 -.531 -.535 -.539 -.5374 -.547 -.547 -.547 -.546 -.551 -.528 -.532 -.536 -.539 -.5383 -.546 -.546 -.547 -.546 -.550 -.528 -.532 -.535 -.539 -.5382 -.547 -.546 -.547 -.547 -.551 -.529 -.533 -.537 -.540 -.539

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace -.089 -.124 -.171 -.228 -.188 -.140 -.086 .008 .104 -.04710 -.347 -.393 -.446 -.502 -.507 -.557 -.623 -.710 -.837 .1049 .102 -.390 -.400 -.455 -.510 -.512 -.558 -.624 -.710 .0088 .031 .106 -.388 -.449 -.465 -.475 -.512 -.558 -.623 -.0867 -.155 .030 .105 -.389 -.450 -.516 -.475 -.512 -.557 -.1406 -.250 -.141 .035 .111 -.387 -.450 -.465 -.510 -.507 -.1885 -.325 -.254 -.143 .034 .111 -.389 -.449 -.455 -.502 -.2284 -.347 -.337 -.245 -.143 .035 .105 -.388 -.400 -.446 -.1713 -.308 -.344 -.337 -.254 -.141 .030 .106 -.390 -.393 -.1242 -.275 -.308 -.347 -.325 -.250 -.155 .031 .102 -.347 -.089


Ace -.479 -.480 -.496 -.537 -.433 -.321 -.224 -.052 .139 -.46810 -.739 -.811 -.899 -1.002 -1.014 -1.113 -1.246 -1.420 -1.674 .1399 .143 -.821 -.819 -.911 -1.024 -1.024 -1.116 -1.249 -1.420 -.0528 .006 .162 -.806 -.910 -.936 -.950 -1.024 -1.116 -1.246 -.2247 -.436 .014 .171 -.791 -.908 -1.035 -.950 -1.024 -1.113 -.3216 -.745 -.445 .011 .171 -.804 -.908 -.936 -1.024 -1.014 -.4335 -.955 -.738 -.438 .018 .171 -.791 -.910 -.911 -1.002 -.5374 -1.073 -.956 -.739 -.438 .011 .171 -.806 -.819 -.899 -.4963 -1.093 -1.071 -.956 -.738 -.445 .014 .162 -.821 -.811 -.4802 -1.094 -1.093 -1.073 -.955 -.745 -.436 .006 .143 -.739 -.479

EV Split2 3 4 5 6 7 8 9 10 Ace

reg -.461 -.508 -.578 -.684 -.651 -.632 -.463 -.266 .060 .194dbs -.453 -.497 -.566 -.672 -.640 -.619 -.453 -.260 .060 .194


6 Decks, S17, Dealer Shows AceEV Stand

2 3 4 5 6 7 8 9 10 Ace

Ace -.666 -.666 -.666 -.666 -.479 -.100 .280 .660 .923 -.66710 -.668 -.668 -.668 -.667 -.665 -.476 -.097 .283 .655 .9239 -.666 -.665 -.665 -.665 -.662 -.663 -.474 -.093 .283 .6608 -.665 -.664 -.664 -.664 -.661 -.662 -.663 -.474 -.097 .2807 -.664 -.664 -.663 -.663 -.660 -.661 -.662 -.663 -.476 -.1006 -.663 -.663 -.662 -.662 -.660 -.660 -.661 -.662 -.665 -.4795 -.665 -.665 -.665 -.664 -.662 -.663 -.664 -.665 -.667 -.6664 -.666 -.665 -.665 -.665 -.662 -.663 -.664 -.665 -.668 -.6663 -.666 -.666 -.665 -.665 -.663 -.664 -.664 -.665 -.668 -.6662 -.666 -.666 -.666 -.665 -.663 -.664 -.665 -.666 -.668 -.666

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace -.060 -.096 -.134 -.172 -.183 -.095 -.008 .079 .145 -.02310 -.350 -.396 -.441 -.483 -.516 -.557 -.628 -.728 -.857 .1459 .147 -.354 -.395 -.440 -.483 -.514 -.555 -.626 -.728 .0798 .082 .147 -.355 -.403 -.442 -.476 -.514 -.555 -.628 -.0087 -.068 .082 .147 -.355 -.405 -.449 -.476 -.514 -.557 -.0956 -.202 -.068 .081 .148 -.357 -.405 -.442 -.483 -.516 -.1835 -.314 -.200 -.066 .083 .148 -.355 -.403 -.440 -.483 -.1724 -.309 -.316 -.199 -.066 .081 .147 -.355 -.395 -.441 -.1343 -.281 -.309 -.316 -.200 -.068 .082 .147 -.354 -.396 -.0962 -.254 -.281 -.309 -.314 -.202 -.068 .082 .147 -.350 -.060


Ace -.620 -.620 -.624 -.629 -.536 -.362 -.189 -.019 .115 -.62010 -.826 -.877 -.932 -.988 -1.032 -1.114 -1.255 -1.455 -1.714 .1159 .121 -.834 -.874 -.930 -.987 -1.027 -1.110 -1.252 -1.455 -.0198 -.008 .124 -.832 -.888 -.933 -.974 -1.027 -1.110 -1.255 -.1897 -.428 -.006 .127 -.830 -.890 -.947 -.974 -1.027 -1.114 -.3626 -.809 -.428 -.004 .130 -.831 -.890 -.933 -.987 -1.032 -.5365 -1.129 -.808 -.427 -.002 .130 -.830 -.888 -.930 -.988 -.6294 -1.303 -1.130 -.808 -.427 -.004 .127 -.832 "-.874 -.932 -.6243 -1.332 -1.303 -1.130 -.808 -.428 -.006 .124 -.834 -.877 -.6202 -1.332 -1.332 -1.303 -1.129 -.809 -.428 -.008 .121 -.826 -.620

EV Split2 3 4 S 6 7 8 9 10 Ace

reg -.410 -.459 -.513 -.571 -.614 -.620 -.386 -.122 .153 .127dbs -.410 -.459 -.513 -.571 -.614 -.620 -.386 -.122 .153 .127

ApPENDIX E 319

6 Decks, HI7, Dealer Shows AceEV Stand

2 3 4 5 6 7 8 9 10 AceAce -.597 -.597 -.597 -.597 -.514 -.225 .190 .606 .905 -.59810 -.599 -.599 -.599 -.599 -.598 -.514 -.223 .192 .600 .9059 -.597 -.597 -.596 -.596 -.596 -.595 -.512 -.220 .192 .6068 -.596 -.596 -.595 -.595 -.595 -.594 -.595 -.512 -.223 .1907 -.595 -.595 -.594 -.594 -.594 -.593 -.594 -.595 -.514 -.2256 -.596 -.595 -.595 -.595 -.595 -.594 -.595 -.596 -.598 -.5145 -.596 -.596 -.596 -.596 -.595 -.594 -.595 -.596 -.599 -.5974 -.596 -.596 -.596 -.596 -.595 -.594 -.595 -.596 -.599 -.5973 -.597 -.596 -.596 -.596 -.595 -.595 -.596 -.597 -.599 -.5972 -.597 -.597 -.596 -.596 -.596 -.595 -.596 -.597 -.599 -.597

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace -.100 -.135 -.172 -.209 -.221 -.161 -.065 .030 .104 -.06510 -.383 -.427 -.469 -.510 -.540 -.579 -.640 -.733 -.858 .1049 .107 -.387 -.425 -.469 -.509 -.539 -.577 -.638 -.733 .0308 .034 .107 -.387 -.433 -.470 -.503 -.539 -.577 -.640 -.0657 -.127 .034 .108 -.388 -.434 -.478 -.503 -.539 -.579 -.1616 -.266 -.125 .035 .109 -.387 -.434 -.470 -.509 -.540 -.221S -.352 -.266 -.124 .034 .109 -.388 -.433 -.469 -.510 -.2094 -.344 -.353 -.264 -.124 .035 .108 -.387 -.425 -.469 -.1723 -.318 -.344 -.353 -.266 -.125 .034 .107 -.387 -.427 -.1352 -.293 -.318 -.344 -.352 -.266 -.127 .034 .107 -.383 -.100


Ace -.584 -.584 -.587 -.593 -.549 -.418 -.228 -.041 .108 -.58510 -.834 -.895 -.959 -1.027 -1.080 -1.158 -1.280 -1.466 -1.717 .1089 .115 -.842 -.891 -.958 -1.026 -1.077 -1.154 -1.277 -1.466 -.0418 -.030 .119 -.839 -.906 -.962 -1.014 -1.0n -1.154 -1.280 -.2287 -.455 -.028 .122 -.837 -.907 -.9n -1.014 -1.077 -1.158 -.4186 -.832 -.453 -.026 .124 -.837 -.907 -.962 -1.026 -1.080 -.5495 -1.086 -.831 -.452 -.026 .124 -.837 -.906 -.958 -1.027 -.5934 -1.180 -1.085 -.830 -.452 -.026 .122 -.839 -.891 -.959 -.5873 -1.193 -1.180 -1.085 -.831 -.453 -.028 .119 -.842 -.895 -.5842 -1.194 -1.193 -1.180 -1.086 -.832 -.455 -.030 .115 -.834 -.584

EV Split2 3 4 S 6 7 8 9 10 Ace

reg -.492 -.538 -.587 -.645 -.682 -.698 -.520 -.240 .055 .121dbs -.490 -.536 -.585 -.642 -.679 -.695 -.518 -.239 .055 .121


6 Decks, S17, Dealer Shows 2EV Stand

2 3 4 5 6 7 8 9 10 Ace

Ace -.291 -.291 -.291 -.289 -.150 .124 .389 .642 .882 -.29010 -.296 -.296 -.296 -.293 -.294 -.154 .121 .386 .638 .8829 -.292 -.292 -.292 -.289 -.290 -.290 -.151 .124 .386 .6428 -.292 -.291 -.291 -.289 -.289 -.290 -.290 -.151 .121 .3897 -.291 -.291 -.291 -.288 -.289 -.289 -.290 -.290 -.154 .1246 -.291 -.291 -.290 -.288 -.288 -.289 -.289 -.290 -.294 -.150S -.290 -.290 -.290 -.287 -.288 -.288 -.289 -.289 -.293 -.2894 -.292 -.292 -.292 -.290 -.290 -.291 -.291 -.292 -.296 -.2913 -.293 -.293 -.292 -.290 -.291 -.291 -.291 -.292 -.296 -.2912 -.293 -.293 -.292 -.290 -.291 -.291 -.292 -.292 -.296 -.291

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .045 .021 -.002 -.023 .001 .063 .123 .183 .240 .08410 -.252 -.307 -.363 -.420 -.470 -.537 -.624 -.732 -.854 .2409 .242 -.256 -.305 -.362 -.419 -.469 -.535 -.623 -.732 .1838 .187 .243 -.254 -.312 -.362 -.413 -.468 -.535 -.624 .1237 .076 .187 .244 -.253 -.312 -.369 -.413 -.469 -.537 .0636 -.020 .077 .188 .246 -.253 -.312 -.362 -.419 -.470 .001S -.108 -.021 .078 .189 .246 -.253 -.312 -.362 -.420 -.0234 -.142 -.111 -.020 .078 .188 .244 -.254 -.305 -.363 -.0023 -.129 -.143 -.111 -.021 .077 .187 .243 -.256 -.307 .0212 -.115 -.129 -.142 -.108 -.020 .076 .187 .242 -.252 .045


Ace -.067 -.068 -.071 -.074 -.004 .121 .241 .362 .473 -.06310 -.504 -.614 -.726 -.839 -.940 -1.073 -1.248 -1.465 -1.708 .4739 .479 -.511 -.611 -.723 -.838 -.937 -1.071 -1.247 -1.465 .3628 .369 .480 -.509 -.623 -.725 -.825 -.937 -1.071 -1.248 .2417 .070 .370 .482 -.507 -.624 -.739 -.825 -.937 -1.073 .1216 -.198 .069 .371 .486 -.507 -.624 -.725 -.838 -.940 -.004S -.431 -.198 .070 .373 .486 -.507 -.623 -.723 -.839 -.0744 -.563 -.434 -.201 .070 .371 .482 -.509 -.611 -.726 -.0713 -.585 -.564 -.434 -.198 .069 .370 .480 -.511 -.614 -.0682 -.586 -.585 -.563 -.431 -.198 .070 .369 .479 -.504 -.067

EV Split2 3 4 5 6 7 8 9 10 Ace

reg -.148 -.201 -.230 -.253 -.273 -.210 -.035 .152 .356 .486dbs -.083 -.135 -.164 -.188 -.206 -.144 .030 .188 .356 .486

ApPENDIX E 321

6 Decks, H17, Dealer Shows 2EV Stand

2 3 4 5 6 7 8 9 10 AceAce -.285 -.285 -.285 -.283 -.153 .113 .381 .638 .880 -.28410 -.290 -.290 -.289 -.287 -.287 -.157 .110 .378 .633 .8809 -.286 -.286 -.286 -.283 -.284 -.284 -.154 .113 .378 .6388 -.285 -.285 -.285 -.282 -.283 -.283 -.284 -.154 .110 .3817 -.285 -.285 -.284 -.282 -.282 -, -.283

~1r.. -.284 -.157 .113

6 -.284 -.284 -.284 -.281 -.282 . ;:.282 -. 83 -.284 -.287 -.1535 -.284 -.284 -.284 -.281 -.281 -.282 -.282 -.283 -.287 -.2834 -.286 -.286 -.286 -.284 -.284 -.284 -.285 -.286 -.289 -.2853 -.286 -.286 -.286 -.284 -.284 -.285 -.285 -.286 -.290 -.2852 -.286 -.286 -.286 -.284 -.284 -.285 -.285 -.286 -.290 -.285

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .046 .022 -.001 -.022 .000 .060 .121 .182 .239 .08210 -.252 -.308 -.364 -.422 -.472 -.538 -.625 -.733 -.854 .2399 .241 -.256 -.306 -.363 -.421 -.471 -.537 -.624 -.733 .1828 .185 .242 -.255 -.312 -.364 -.414 -.471 -.537 -.625 .1217 .074 .186 .243 -.254 -.313 -.371 -.414 -.471 -.538 .0606 -.023 .075 .187 .245 -.254 -.313 -.364 -.421 -.472 .0005 -.108 -.023 .076 .188 .245 -.254 -.312 -.363 -.422 -.0224 -.140 -.111 -.022 .076 .187 .243 -.255 -.306 -.364 -.0013 -.126 -.140 -.111 -.023 .075 .186 .242 -.256 -.308 .0222 -.113 -.126 -.140 -.108 -.023 .074 .185 .241 -.252 .046


Ace -.064 -.064 -.068 -.070 -.005 .116 .237 .360 .472 -.06010 -.505 -.616 -.729 -.843 -.945 -.1.077 -1.250 -1.466 -1.708 .4729 .478 -.512 -.612 -.726 -.841 -.942 1.075 -1.249 -1.466 .3608 .367 .480 -.510 -.625 -.727 -.829 -.941 -1.075 -1.250 .2377 .068 .368 .482 -.507 -.625 -.741 -.829 -.942 -1.0n .1166 -.200 .067 .369 .485 -.507 -.625 -.727 -.841 -.945 -.0055 -.427 -.200 .068 .371 .485 -.507 -.625 -.726 -.843 -.0704 -.552 -.430 -.202 .068 .369 .482 -.510 -.612 -.729 -.0683 -.573 -.552 -.430 -.200 .067 .368 .480 -.512 -.616 -.0642 -.573 -.573 -.552 -.427 -.200 .068 .367 .478 -.505 -.064

EV Split2 3 4 5 6 7 8 9 10 Ace

reg -.149 -.197 -.226 -.248 -.267 -.211 -.040 .148 .353 .485dbs -.083 -.132 -.160 -.183 -.200 -.145 .024 .183 .353 .485



2 3 4 5 6 7 8 9 10 Ace

Ace -.250 -.250 -.248 -.247 -.114 .151 .407 .649 .885 -.24910 -.255 -.254 -.252 -.252 -.252 -.118 .148 .401 .648 .8859 -.254 -.254 -.251 -.251 -.251 -.252 -.118 .145 .401 .6498 -.250 -.250 -.247 -.247 -.247 -.248 -.249 -.118 .148 .4077 -.249 -.249 -.247 -.246 -.247 -.247 -.248 -.252 -.118 .1516 -.249 -.249 -.246 -.246 -.246 -.247 -.247 -.251 -.252 -.1145 -.248 -.248 -.246 -.245 -.246 -.246 -.247 -.251 -.252 -.2474 -.249 -.249 -.246 -.246 -.246 -.247 -.247 -.251 -.252 -.2483 -.251 -.251 -.249 -.248 -.249 -.249 -.250 -.254 -.254 -,2502 -.251 -.251 -.249 -.248 -.249 -.249 -.250 -.254 -.255 -,250

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .074 .050 .028 .007 .030 .091 .153 .205 .262 ,10610 -.231 -.290 -.350 -.410 -.463 -.532 -.622 -.726 -.854 .2629 .262 -.237 -.289 -.350 -.410 -.463 -.532 -.623 -.726 .2058 .212 .266 -.233 -.294 -.349 -.402 -.462 -.532 -.622 .1537 .106 .212 .267 -.232 -.293 -.355 -.402 -.463 -.532 ,0916 .010 .107 .213 .269 -.232 -.293 -.349 -.410 -.463 .0305 -.074 .010 .108 .214 .269 -.232 -.294 -.350 -.410 ,0074 -.107 -.077 .011 .108 .213 .267 -.233 -.289 -.350 ,0283 -.096 -.109 -.077 .010 .107 .212 .266 -.237 -.290 ,0502 -.082 -.096 -.107 -.074 .010 .106 .212 .262 -.231 .074


Ace -.001 -.004 -.006 -.009 .058 .179 .303 .407 .521 .00310 -.463 -.580 -.700 -.820 -.927 -1.065 -1.244 -1.451 -1.707 .5219 .521 -.474 -.579 -.700 -.821 -.926 -1.065 -1.246 -1.451 .4078 .421 .529 -.466 -.587 -.697 -.805 -.923 -1.065 -1.244 .3037 .132 .422 .532 -.464 -.587 -.710 -.805 -.926 -1.065 .1796 -.127 .132 .424 .535 -.464 -.587 -.697 -.821 -.927 .0585 -.352 -.127 .133 .425 .535 -.464 -.587 -.700 -.820 -.0094 -.477 -.353 -.127 .133 .424 .532 -.466 -.579 -.700 -.0063 -.502 -.481 -.353 -.127 .132 .422 .529 -.474 -.580 -.0042 -.503 -.502 -.477 -.352 -.127 .132 .421 .521 -,463 -.001

EV Split2 3 4 5 6 7 8 9 10 Ace

reg -.095 -.136 -.159 -.184 -.203 -.143 .026 .198 ,404 .533dbs -.022 -.062 -.084 -.112 -.128 -.069 .098 .238 .404 .533

ApPENDIX E 323


2 3 4 5 6 7 8 9 10 AceAce -.245 -.245 -.242 -.242 -.116 .142 .400 .645 .884 -.24410 -.249 -.249 -.246 -.246 -.247 -.121 .138 .394 .644 .8849 -.248 -.248 -.246 -.245 -.246 -.246 -.121 .135 .394 .6458 -.244 -.244 -.242 -.241 -.242 -.242 -.243 -.121 .138 .4007 -.244 -.244 -.241 -.241 -.241 -.242 -.242 -.246 -.121 .1426 -.243 -.243 -.241 -.240 -.241 -.241 -.242 -.246 -.247 -.1165 -.243 -.243 -.240 -.240 -.240 -.241 -.241 -.245 -.246 -.2424 -.243 -.243 -.241 -.240 -.241 -.241 -.242 -.246 -.246 -.2423 -.246 -.246 -.243 -.243 -.243 -.244 -.244 -.248 -.249 -.2452 -.246 -.246 -.243 -.243 -.243 -.244 -.244 -.248 -.249 -.245

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .074 .050 .029 .008 .029 .088 .151 .203 .261 .10410 -.232 -.291 -.351 -.411 -.465 -.534 -.623 -.726 -.854 .2619 .261 -.237 -.290 -.351 -.412 -.465 -.534 -.624 -.726 .2038 .210 .265 -.233 -.294 -.350 -.404 -.464 -.534 -.623 .1517 .104 .211 .267 -.232 -.294 -.356 -.404 -.465 -.534 .0886 .008 .105 .212 .268 -.232 -.294 -.350 -.412 -.465 .0295 -.074 .008 .106 .212 .268 -.232 -.294 -.351 -.411 .0084 -.104 -.077 .009 .106 .212 .267 -.233 -.290 -.351 .0293 -.093 -.106 -.077 .008 .105 .211 .265 -.237 -.291 .0502 -.080 -.093 -.104 -.074 .008 .104 .210 .261 -.232 .074


Ace .002 -.001 -.003 -.006 .057 .175 .300 .405 .520 .00510 -.464 -.581 -.702 -.823 -.931 -1.068 -1.246 -1.452 -1.707 .5209 .521 -.475 -.580 -.702 -.824 -.930 -1.068 -1.248 -1.452 .4058 .419 .529 -.467 -.589 -.699 -.808 -.927 -1.068 -1.246 .3007 .130 .421 .532 -.465 -.588 -.713 -.808 -.930 -1.068 .1756 -.129 .130 .422 .535 -.465 -.588 -.699 -.824 -.931 .0575 -.348 -.129 .131 .424 .535 -.465 -.589 -.702 -.823 -.0064 -.467 -.350 -.129 .131 .422 .532 -.467 -.580 -.702 -.0033 -.492 -.472 -.350 -.129 .130 .421 .529 -.475 -.581 -.0012 -.492 -.492 -.467 -.348 -.129 .130 .419 .521 -.464 .002

EV Split2 3 4 5 6 7 8 9 10 Ace

reg -.095 -.133 -.155 -.180 -.198 -.143 .021 .194 .402 .533dbs -.022 -.059 -.080 -.107 -.123 -.070 .093 .234 .402 .533



2 3 4 5 6 7 8 9 10 Ace

Ace -.207 -.205 -.204 -.204 -.074 .180 .422 .660 .888 -.20610 -.211 -.209 -.209 -.208 -.209 -.078 .174 .420 .658 .8889 -.211 -.208 -.208 -.207 -.208 -.209 -.081 .174 .420 .6608 -.210 -.207 -.207 -.207 -.207 -.208 -.212 -.081 .174 .4227 -.206 -.203 -.203 -.203 -.203 -.204 -.208 -.209 -.078 .1806 -.205 -.203 -.202 -.202 -.202 -.203 -.207 -.208 -.209 -.074S -.205 -.202 -.202 -.201 -.202 -.203 -.207 -.207 -.208 -.2044 -.205 -.203 -.202 -.202 -.202 -.203 -.207 -.208 -.209 -.2053 -.205 -.203 -.203 -.202 -.203 -.203 -.207 -.208 -.209 -.2052 -.208 -.205 -.205 -.205 -.205 -.206 -.210 -.211 -.211 -.207

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .104 .082 .059 .039 .062 .124 .177 .230 .285 .12910 -.210 -.272 -.335 -.399 -.456 -.528 -.614 -.725 -.853 .2859 .286 -.215 -.271 -.335 -.400 -.456 -.529 -.614 -.725 .2308 .234 .288 -.214 -.277 -.335 -.393 -.457 -.529 -.614 .1777 .135 .239 .292 -.210 -.275 -.341 -.393 -.456 -.528 .1246 .045 .138 .240 .294 -.210 -.275 -.335 -.400 -.456 .062S -.038 .046 .138 .240 .294 -.210 -.2n -.335 -.399 .0394 -.070 -.039 .048 .138 .240 .292 -.214 -.271 -.335 .0593 -.058 -.069 -.039 .046 .138 .239 .288 -.215 -.272 .0822 -.047 -.058 -.070 -.038 .045 .135 .234 .286 -.210 .104


Ace .068 .067 .062 .058 .124 .249 .355 .460 .571 ,07110 -.421 -.543 -.670 -.799 -.912 -1.057 -1.228 -1.449 -1.707 ,5719 .571 -.431 -.542 -.670 -.799 -.912 -1.058 -1.228 -1.449 .4608 .468 .576 -.427 -.554 -.671 -.786 -.914 -1.058 -1.228 ,3557 .196 .478 .584 -.421 -.550 -.681 -.786 -.912 -1.057 ,2496 -.053 .199 .480 .587 -.420 -.550 -.671 -.799 -.912 .124S -.269 -.050 .200 .481 .587 -.421 -.554 -.670 -.799 .0584 -.390 -.266 -.050 .200 .480 .584 -.427 -.542 -.670 .0623 -.411 -.387 -.266 -.050 .199 .478 .576 -.431 -.543 .0672 -.416 -.411 -.390 -.269 -.053 .196 .468 .571 -.421 ,068

EV Split2 J 4 5 6 7 8 9 10 Ace

reg -.038 -.060 -.086 -.110 -.128 -.070 .082 .257 .454 ,583dbs .048 .030 .008 -.017 -.025 .033 .163 .301 .454 ,583

ApPENDIX E 325

6 Decks, HI7, Dealer Shows 4EV Stand

2 J 4 5 6 7 8 9 10 AceAce -.202 -.199 -.199 -.199 -.076 .171 .416 .656 .887 -.20110 -.206 -.204 -.203 -.203 -.203 -.081 .164 .413 .654 .8879 -.205 -.203 -.202 -.202 -.203 -.203 -.084 .164 .413 .6568 -.205 -.202 -.202 -.201 -.202 -.202 -.206 -.084 .164 .4167 -.201 -.198 -.198 -.197 -.198 -.198 -.202 -.203 -.081 .1716 -.200 -.197 -.197 -.197 -.197 -.198 -.202 -.203 -.203 -.076S -.199 -.197 -.196 -.196 -.197 -.197 -.201 -.202 -.203 -.1994 -.200 -.197 -.197 -.196 -.197 -.198 -.202 -.202 -.203 -.199J -.200 -.198 -.197 -.197 -.197 -.198 -.202 -.203 -.204 -.1992 -.203 -.200 -.200 -.199 -.200 -.201 -.205 -.205 -.206 -.202

EV Hit2 J 4 5 6 7 8 9 10 Ace

Ace .104 .083 .061 .041 .062 .122 .176 .229 .285 .12910 -.211 -.272 -.336 -.401 -.458 -.530 -.615 -.725 -.853 .2859 .285 -.216 -.272 -.336 -.401 -.458 -.531 -.615 -.725 .2298 .233 .288 -.214 -.278 -.337 -.395 -.459 -.531 -.615 .1767 .134 .238 .292 -.211 -.276 -.342 -.395 -.458 -.530 .1226 .044 .137 .239 .293 -.210 -.276 -.337 -.401 -.458 .062S -.038 .044 .137 .240 .293 -.211 -.278 -.336 -.401 .0414 -.067 -.039 .046 .137 .239 .292 -.214 -.272 -.336 .061J -.056 -.066 -.039 .044 .137 .238 .288 -.216 -.272 .0832 -.045 -.056 -.067 -.038 .044 .134 .233 .285 -.211 .104

EV Double Down2 J 4 5 6 7 8 9 10 Ace

Ace .070 .069 .065 .061 .123 .245 .352 .459 .570 .07310 -.421 -.545 -.673 -.801 -.916 -1.060 -1.230 -1.450 -1.707 .5709 .571 -.432 -.544 -.672 -.802 -.915 -1.062 -1.230 -1.450 .4598 .467 .575 -.428 -.556 -.673 -.789 -.918 -1.062 -1.230 .3527 .194 .477 .584 -.421 -.551 -.683 -.789 -.915 -1.060 .2456 -.054 .197 .479 .587 -.421 -.551 -.673 -.802 -.916 .123S -.265 -.051 .198 .479 .587 -.421 -.556 -.672 -.801 .0614 -.381 -.263 -.051 .198 .479 .584 -.428 -.544 -.673 .065J -.401 -.3n -.263 -.051 .197 .4n .575 -.432 -.545 .0692 -.406 -.401 -.381 -.265 -.054 .194 .467 .571 -.421 .070

EV Split2 3 4 5 6 7 8 9 10 Ace

reg -.034 -.055 -.080 -.105 -.122 -.070 .079 .254 .452 .582dbs .052 .035 .014 -.011 -.020 .034 .159 .298 .452 .582



2 3 4 5 6 7 8 9 10 Ace

Ace -.160 -.159 -.159 -.158 -.037 .203 .443 .672 .892 -.16110 -.164 -.163 -.163 -.163 -.163 -.045 .200 .441 .671 .8929 -.163 -.163 -.162 -.162 -.163 -.167 -.045 .200 .441 .6728 -.162 -.162 -.162 -.161 -.162 -.166 -.167 -.045 .200 .4437 -.162 -.161 -.161 -.160 -.161 -.165 -.166 -.167 -.045 .2036 -.158 -.157 -.157 -.156 -.157 -.161 -.162 -.163 -.163 -.0375 -.157 -.156 -.156 -.156 -.156 -.160 -.161 -.162 -.163 -.1584 -.157 -.157 -.156 -.156 -.157 -.161 -.162 -.162 -.163 -.1593 -.158 -.157 -.157 -.156 -.157 -.161 -.162 -.163 -.163 -.1592 -.158 -.158 -.157 -.157 -.158 -.162 -.162 -.163 -.164 -.160

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .137 .116 .094 .075 .099 .152 .207 .258 .311 .16110 -.189 -.254 -.320 -.388 -.449 -.518 -.611 -.723 -.853 .3119 .312 -.194 -.253 -.320 -.388 -.450 -.518 -.611 -.723 .2588 .263 .314 -.192 -.259 -.321 -.383 -.450 -.518 -.611 .2077 .164 .264 .316 -.191 -.259 -.329 -.383 -.450 -.518 .1526 .080 .170 .269 .321 -.189 -.259 -.321 -.388 -.449 .0995 .002 .082 .172 .270 .321 -.191 -.259 -.320 -.388 .0754 -.028 .001 .084 .172 .269 .316 -.192 -.253 -.320 .0943 -.017 -.028 .001 .082 .170 .264 .314 -.194 -.254 .1162 -.005 -.017 -.028 .002 .080 .164 .263 .312 -.189 .137


Ace .140 .138 .134 .129 .198 .304 .414 .516 .622 .14110 -.377 -.508 -.641 -.776 -.898 -1.036 -1.223 -1.447 -1.706 .6229 .625 -.388 -.507 -.641 -.777 -.900 -1.036 -1.223 -1.447 .5168 .525 .628 -.385 -.519 -.642 -.767 -.900 -1.036 -1.223 .4147 .257 .529 .632 -.382 -.519 -.658 -.767 -.900 -1.036 .3046 .023 .266 .538 .641 -.377 -.519 -.642 -.777 -.898 .1985 -.183 .026 .269 .540 .641 -.382 -.519 -.641 -.776 .1294 -.296 -.181 .028 .269 .538 .632 -.385 -.507 -.641 .1343 -.315 -.296 -.181 .026 .266 .529 .628 -.388 -.508 .1382 -.316 -.315 -.296 -.183 .023 .257 .525 .625 -.377 .140

EV Split2 3 4 5 6 7 8 9 10 Ace

reg .043 .018 -.007 -.031 -.048 -.006 .150 .319 .510 .634dbs .149 .128 .107 .082 .074 .115 .239 .367 .510 .634

ApPENDIX E 327


2 3 4 5 6 7 8 9 10 Ace

Ace -.157 -.157 -.156 -.156 -.039 .199 .440 .670 .891 -.15910 -.161 -.161 -.160 -.160 -.161 -.046 .195 .437 .669 .8919 -.161 -.160 -.160 -.159 -.160 -.164 -.046 .195 .437 .6708 -.160 -.159 -.159 -.159 -.159 -.163 -.164 -.046 .195 .4407 -.159 -.159 -.158 -.158 -.159 -.163 -.163 -.164 -.046 .1996 -.155 -.155 -.154 -.154 -.155 -.159 -.159 -.160 -.161 -.0395 -.154 -.154 -.154 -.153 -.154 -.158 -.159 -.159 -.160 -.1564 -.155 -.154 -.154 -.154 -.154 -.158 -.159 -.160 -.160 -.1563 -.155 -.155 -.154 -.154 -.155 -.159 -.159 -.160 -.161 -.1572 -.156 -.155 -.155 -.154 -.155 -.159 -.160 -.161 -.161 -.157

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .138 .116 .095 .075 .099 .151 .206 .258 .311 .16110 -.189 -.254 -.321 -.389 -.450 -.519 -.612 -.723 -.853 .3119 .312 -.194 -.254 -.321 -.389 -.451 -.519 -.612 -.723 .2588 .262 .314 -.192 -260 -.321 -.384 -.451 -.519 -.612 .2067 .163 .264 .316 -.191 -.260 -.330 -.384 -.451 -.519 .1516 .079 .169 .269 .320 -.189 -.260 -.321 -.389 -.450 .0995 .003 .081 .171 .270 .320 -.191 -.260 -.321 -.389 .0754 -.027 .001 .083 .171 .269 .316 -.192 -.254 -.321 .0953 -.016 -.027 .001 .081 .169 .264 .314 -.194 -.254 .1162 -.004 -.016 -.027 .003 .079 .163 .262 .312 -.189 .138


Ace .141 .139 .135 .130 .197 .302 .412 .515 .622 .14210 -.378 -.509 -.642 -.777 -.899 -1.038 -1.224 -1.447 -1.706 .6229 .624 -.388 -.508 -.642 -.778 -.902 -1.038 -1.224 -1.447 .5158 .524 .628 -.385 -.520 -.643 -.768 -.902 -1.038 -1.224 .4127 .256 .528 .632 -.382 -.519 -.659 -.768 -.902 -1.038 .3026 .022 .265 .537 .641 -.377 -.519 -.643 -.778 -.899 .1975 -.181 .026 .268 .540 .641 -.382 -.520 -.642 -.777 .1304 -.291 -.180 .027 .268 .537 .632 -.385 -.508 -.642 .1353 -.310 -.291 -.180 .026 .265 .528 .628 -.388 -.509 .1392 -.311 -.310 -.291 -.181 .022 .256 .524 .624 -.378 .141

EV Split2 3 4 5 6 7 8 9 10 Ace

reg .045 .020 -.004 -.028 -.046 -.005 .148 .318 .509 .634dbs .150 .130 .109 .084 .076 .116 .237 .366 .509 .634



2 3 4 5 6 7 8 9 10 Ace

Ace -.147 -.147 -.147 -.146 .012 .280 .494 .703 .902 -.14610 -.155 -.154 -.154 -.154 -.158 .008 .281 .494 .703 .9029 -.154 -.154 -.153 -.153 -.157 -.158 .008 .281 .494 .7038 -.153 -.153 -.153 -.152 -.156 -.157 -.158 .008 .282 .4947 -.153 -.152 -.152 -.152 -.156 -.156 -.157 -.158 .008 .2806 -.152 -.152 -.151 -.151 -.155 -.156 -.156 -.157 -.158 .0125 -.148 -.148 -.147 -.147 -.151 -.152 -.152 -.153 -.154 -.1464 -.148 -.148 -.148 -.147 -.151 -.152 -.153 -.153 -.154 -,1473 -.149 -.148 -.148 -.148 -.152 -.152 -.153 -.154 -.154 -.1472 -.149 -.149 -.148 -.148 -.152 -.153 -.153 -.154 -.155 -.147

EV Hit2 3 4 S 6 7 8 9 10 Ace

Ace .163 .140 .118 .101 .129 .191 .240 .287 .335 .18810 -.169 -.235 -.302 -.369 -.425 -.505 -.604 -.720 -.852 .3359 .336 -.174 -.234 -.302 -.371 -.425 -.505 -.604 -.720 .2878 .291 .337 -.173 -.241 -.304 -.365 -.425 -.505 -.604 .2407 .199 .293 .339 -.172 -.243 -.311 -.365 -.425 -.505 .1916 .118 .202 .295 .341 -.174 -.243 -.304 -.371 -.425 .1295 .036 .123 .206 .299 .341 -.172 -.241 -.302 -.369 .1014 -.009 .034 .124 .206 .295 .339 -.173 -.234 -.302 .1183 .002 -.009 .034 .123 .202 .293 .337 -.174 -.235 .1402 .014 .002 -.009 .036 .118 .199 .291 .336 -.169 .163


Ace .188 .186 .183 .185 .258 .383 .480 .573 .670 .19110 -.338 -.469 -.604 -.739 -.851 -1.009 -1.208 -1.440 -1.705 .6709 .672 -.348 -.469 -.604 -.743 -.851 -1.009 -1.208 -1.440 ,5738 .582 .675 -.345 -.481 -.608 -.730 -.851 -1.009 -1.208 .4807 .324 .586 .678 -.343 -.485 -.622 -.730 -.851 -1.009 ,3836 .093 .327 .590 .683 -.348 -.485 -.608 -.743 -.851 ,2585 -.126 .103 .337 .599 .683 -.343 -.481 -.604 -.739 .1854 -.272 -.126 .103 .337 .590 .678 -.345 -.469 -.604 .1833 -.297 -.271 -.126 .103 .327 .586 .675 -.348 -.469 .1862 -.298 -.297 -.272 -.126 .093 .324 .582 .672 -.338 .188

EV Split2 3 4 S 6 7 8 9 10 Ace

reg .086 .059 .032 .007 -.022 .059 .230 .387 .567 .682dbs .208 .184 .161 .136 .116 .193 .328 .439 .567 .682

ApPENDIX E 329

6 Decks, HI7, Dealer Shows 6EV Stand

2 3 4 5 6 7 8 9 10 Ace

Ace -.115 -.115 -.115 -.115 -.005 .223 .452 .678 .894 -.11510 -.122 -.121 -.121 -.121 -.125 -.010 .220 .451 .677 .8949 -.121 -.121 -.120 -.120 -.124 -.125 -.010 .220 .451 .6788 -.120 -.120 -.120 -.120 -.123 -.124 -.125 -.010 .220 .4527 -.120 -.119 -.119 -.119 -.123 -.124 -.124 -.125 -.010 .2236 -.119 -.118 -.118 -.118 -.122 -.123 -.123 -.124 -.125 -.0055 -.115 -.114 -.114 -.114 -.118 -.119 -.120 -.120 -.121 -.1154 -.115 -.115 -.114 -.114 -.118 -.119 -.120 -.120 -.121 -.1153 -.115 -.115 -.115 -.114 -.118 -.119 -.120 -.121 -.121 -.1152 -.116 -.115 -.115 -.115 -.119 -.120 -.120 -.121 -.122 -.115

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .167 .146 .126 .111 .126 .178 .231 .281 .333 .19010 -.171 -.239 -.309 -.379 -.438 -.515 -.610 -.723 -.853 .3339 .334 -.176 -.239 -.309 -.381 -.438 -.515 -.610 -.723 .2818 .286 .336 -.174 -.245 -.311 -.375 -.438 -.515 -.610 .2317 .190 .287 .338 -.173 -.247 -.318 -.375 -.438 -.515 .1786 .107 .193 .289 .340 -.175 -.247 -.311 -.381 -.438 .126S .037 .111 .198 .294 .340 -.173 -.245 -.309 -.379 .1114 .010 .036 .113 .198 .289 .338 -.174 -.239 -.309 .1263 .019 .010 .036 .111 .193 .287 .336 -.176 -.239 .1462 .030 .019 .010 .037 .107 .190 .286 .334 -.171 .167


Ace .204 .203 .200 .202 .252 .357 .462 .563 .667 .20610 -.341 -.478 -.617 -.758 -.875 -1.030 -1.220 -1.445 -1.706 .6679 .669 -.352 -.4n -.618 -.762 -.875 -1.031 -1.220 -1.445 .5638 .571 .672 -.349 -.490 -.622 -.750 -.875 -1.031 -1.220 .4627 .311 .575 .676 -.346 -.493 -.636 -.750 -.875 -1.030 .3576 .082 .315 .579 .680 -.350 -.493 -.622 -.762 -.875 .2525 -.105 .092 .324 .587 .680 -.346 -.490 -.618 -.758 .2024 -.212 -.105 .093 .324 .579 .676 -.349 -.477 -.617 .2003 -.231 -.212 -.105 .092 .315 .575 .672 -.352 -.478 .2032 -.231 -.231 -.212 -.105 .082 .311 .571 .669 -.341 .204

EV Split2 3 4 5 6 7 8 9 10 Ace

reg .111 .087 .064 .041 .013 .061 .207 .370 .556 .680dbs .234 .213 .193 .169 .150 .198 .303 .421 .556 .680


6 Decks, Dealer Shows 7EV Stand

2 3 4 5 6 7 8 9 10 Ace

Ace -.473 -.473 -.473 -.474 -.104 .402 .616 .773 .926 -.47210 -.475 -.475 -.474 -.476 -.477 -.109 .398 .615 .772 .9269 -.478 -.478 -.478 -.479 -.480 -.481 -.110 .400 .615 .7738 -.477 -.4n -.477 -.478 -.479 -.480 -.480 -.110 .398 .6167 -.4n -.477 -.476 -.478 -.478 -.479 -.480 -.481 -.109 .4026 -.476 -.476 -.476 -.477 -.478 -.478 -.479 -.480 -.477 -.1045 -.475 -.475 -.475 -.476 -.477 -.478 -.478 -.479 -.476 -.4744 -.474 -.474 -.473 -.475 -.476 -.476 -.477 -.478 -.474 -.4733 -.474 -.474 -.474 -.475 -.476 -.477 -.477 -.478 -.475 -.4732 -.475 -.474 -.474 -.475 -.476 -.477 -.477 -.478 -.475 -.473

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .120 .077 .036 -.008 .055 .172 .221 .255 .291 .16410 -.213 -.269 -.325 -.369 -.409 -.478 -.587 -.713 -.850 .2919 .292 -.218 -.270 -.325 -.369 -.408 -.478 -.587 -.713 .2558 .259 .292 -.218 -.278 -.325 -.363 -.408 -.478 -.587 .2217 .174 .260 .293 -.220 -.278 -.332 -.363 -.408 -.478 .1726 .084 .176 .262 .293 -.221 -.278 -.325 -.369 -.409 .0555 -.068 .084 .176 .260 .293 -.220 -.278 -.325 -.369 -.0084 -.153 -.069 .087 .176 .262 .293 -.218 -.270 -.325 .0363 -.119 -.154 -.069 .084 .176 .260 .292 -.218 -.269 .0772 -.089 -.119 -.153 -.068 .084 .174 .259 .292 -.213 .120


Ace -.180 -.182 -.177 -.184 -.009 .224 .321 .386 .464 -.17610 -.505 -.587 -.674 -.747 -.817 -.957 -1.175 -1.426 -1.701 .4649 .462 -.518 -.589 -.677 -.748 -.817 -.956 -1.174 -1.426 .3868 .396 .464 -.516 -.606 -.678 -.736 -.817 -.956 -1.175 .3217 .114 .401 .468 -.518 -.607 -.692 -.736 -.817 -.957 .2246 -.181 .116 .406 .470 -.520 -.607 -.678 -.748 -.817 -.009S -.586 -.179 .118 .405 .470 -.518 -.606 -.677 -.747 -.1844 -.890 -.582 -.175 .118 .406 .468 -.516 -.589 -.674 -.1773 -.949 -.890 -.582 -.179 .116 .401 .464 -.518 -.587 -.1822 -.949 -.949 -.890 -.586 -.181 .114 .396 .462 -.505 -.180

EV Split2 3 4 5 6 7 8 9 10 Ace

reg -.054 -.115 -.183 -.248 -.299 -.135 .171 .343 .509 .476dbs -.005 -.066 -.133 -.199 -.249 -.086 .218 .369 .509 .476

ApPENDIX E 331


2 3 4 5 6 7 8 9 10 AceAce -.510 -.510 -.511 -.511 -.383 .108 .596 .791 .930 -.50910 -.512 -.511 -.513 -.513 -.513 -.384 .104 .591 .790 .9309 -.511 -.511 -.512 -.512 -.513 -.513 -.388 .099 .591 .7918 -.514 -.514 -.516 -.515 -.516 -.517 -.518 -.388 .104 .5967 -.514 -.513 -.515 -.515 -.515 -.516 -.517 -.513 -.384 .1086 -.513 -.513 -.514 -.514 -.515 -.515 -.516 -.513 -.513 -.3835 -.512 -.512 -.513 -.513 -.514 -.515 -.515 -.512 -.513 -.5114 -.512 -.512 -.513 -.513 -.514 -.515 -.516 -.512 -.513 -.5113 -.511 -.511 -.512 -.512 -.513 -.513 -.514 -.511 -.511 -.5102 -.511 -.511 -.512 -.512 -.513 -.514 -.514 -.511 -.512 -.510

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace .052 .017 -.028 -,069 -.072 .041 .153 .194 .229 .09510 -.272 -.234 -.370 -.417 -.453 -.501 -.587 -.711 -.850 .2299 .228 -.279 -.326 -.372 -.417 -.454 -.501 -.587 -.711 .1948 .199 .229 -.279 -.333 -.371 -.411 -.453 -.501 -.587 .1537 .100 .201 .229 -.279 -.335 -.378 -.411 -.454 -.501 .0416 -.059 .101 .200 .230 -.279 -.335 -.371 -.417 -.453 -.0725 -.212 -.059 .100 .200 .230 -.279 -.333 -.372 -.417 -.0694 -.220 -.213 -.059 .100 .200 .229 -.279 -.326 -.370 -.0283 -.187 -.219 -.213 -.059 .101 .201 .229 -.279 -.324 .0172 -.157 -.187 -.220 -.212 -.059 .100 .199 .228 -.272 .052


Ace -.314 -.305 -.314 -.317 -.251 -.028 .194 .277 .348 -.31210 -.617 -.693 -.762 -.842 -.906 -1.002 -1.174 -1.422 -1.700 .3489 .347 -.630 -.696 -.765 -.844 -.907 -1.002 -1.174 -1.422 .2n8 .288 .347 -.632 -.713 -.768 -.831 -.907 -1.002 -1.174 .1947 -.022 .293 .349 -.631 -.715 -.781 -.831 -.907 -1.002 -.0286 -.450 -.020 .292 .353 -.631 -.715 -.768 -.844 -.906 -.2515 -.848 -.448 -.022 .293 .353 -.631 -.713 -.765 -.842 -.3174 -1.005 -.847 -.451 -.022 .292 .349 -.632 -.696 -.762 -.3143 -1.022 -1.001 -.847 -.448 -.020 .293 .347 -.630 -.693 -.3052 -1.023 -1.022 -1.005 -.848 -.450 -.022 .288 .347 -.617 -.314

EV Split2 3 4 5 6 7 8 9 10 Ace

reg -.207 -.263 -.322 -.387 -.432 -.422 -.116 .193 .389 .360dbs -.174 -.230 -.288 -.354 -.398 -.389 -.085 .212 .389 .360



2 3 4 5 6 7 8 9 10 Ace

Ace -.539 -.541 -.540 -.540 -.421 -.183 .288 .759 .939 -.53810 -.541 -.542 -.542 -.542 -.543 -.422 -.185 .284 .756 .9399 -.540 -.542 -.541 -.541 -.542 -.543 -.421 -.185 .284 .7598 -.540 -.541 -.541 -.541 -.541 -.542 -.539 . -.421 -.185 .2887 -.543 -.544 -.544 -.544 -.545 -.546 -.542 -.543 -.422 -.1836 -.542 -.543 -.543 -.543 -.544 -.545 -.541 -.542 -.543 -.4215 -.541 -.543 -.543 -.543 -.543 -.544 -.541 -.541 -.542 -,5404 -.542 -.543 -.543 -.543 -.543 -.544 -.541 -.541 -.542 -,5403 -.542 -.543 -.543 -.543 -.543 -.544 -.541 -.542 -.542 -.5412 -.540 -.542 -.542 -.541 -.542 -.543 -.540 -.540 -.541 -.539

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace -.034 -.073 -.112 -.151 -.147 -.098 .007 .113 .157 .00010 -.341 -.383 -.428 -.472 -.505 -.549 -.613 -.713 -.849 .1579 .156 -.348 -.384 -.430 -.473 -.505 -.550 -.613 -.713 .1138 .117 .155 -.349 -.391 -.432 -.467 -.506 -.550 -.613 .0077 -.052 .117 .157 -.348 -.392 -.438 -.467 -.505 -.549 -.0986 -.210 -.052 .117 .157 -.347 -.392 -.432 -.473 -.505 -.1475 -.285 -.211 -.052 .117 .157 -.348 -.391 -.430 -.472 -.1514 -.294 -.288 -.210 -.052 .117 .157 -.349 -.384 -.428 -.1123 -.266 -.295 -.288 -.211 -.052 .117 .155 -.348 -.383 -.0732 -.238 -.266 -.294 -.285 -.210 -.052 .117 .156 -.341 -.034


Ace -.443 -.446 -.451 -.456 -.392 -.285 -.071 .139 .226 -.45110 -.739 -.799 -.872 -.949 -1.009 -1.099 -1.226 -1.426 -1.699 .2269 .225 -.754 -.800 -.875 -.951 -1.010 -1.100 -1.226 -1.426 .1398 .149 .226 -.753 -.814 -.879 -.940 -1.011 -1.100 -1.226 -.0717 -.297 .146 .227 -.752 -.818 -.894 -.940 -1.010 -1.099 -.2856 -.714 -.299 .148 .230 -.750 -.818 -.879 -.951 -1.009 -.3925 -.952 -.716 -.297 .149 .230 -.752 -.814 -.875 -.949 -.4564 -1.064 -.955 -.716 -.297 .148 .227 -.753 -.800 -.872 -.4513 -1.083 -1.067 -.955 -.716 -.299 .146 .226 -.754 -.799 -.4462 -1.080 -1.083 -1.064 -.952 -.714 -.297 .149 .225 -.739 -.443

EV Split2 3 4 S 6 7 8 9 10 Ace

reg -.380 -.429 -.483 -.544 -.583 -.570 -.421 -.106 .225 .238dbs -.364 -.413 -.467 -.528 -.567 -.555 -.405 -.095 .225 .238

ApPENDIX E 333


2 3 4 5 6 7 8 9 10 AceAce -.540 -.540 -.540 -.540 -.420 -.180 .063 .555 .962 -.53910 -.540 -.540 -.540 -.540 -.541 -.418 -.175 .069 .559 .9629 -.540 -.540 -.540 -.540 -.540 -.537 -.415 -.171 .069 .5558 -.539 -.539 -.539 -.539 -.540 -.536 -.537 -.415 -.175 .0637 -.539 -.539 -.538 -.538 -.539 -.536 -.536 -.537 -.418 -.1806 -.542 -.542 -.542 -.542 -.543 -.539 -.540 -.540 -.541 -.4205 -.541 -.541 -.541 -.541 -.542 -.538 -.539 -.540 -.540 -.5404 -.541 -.541 -.541 -,541 -.542 -.538 -.539 -.540 -.540 -.5403 -.542 -.541 -.541 -.541 -.542 -.539 -.539 -.540 -.540 -.5402 -.542 -.542 -.541 -.541 -.542 -.539 -.539 -.540 -.540 -.540

EV Hit2 3 4 5 6 7 8 9 10 Ace

Ace -.102 -.137 -.173 -.211 -.196 -.143 -.088 .023 .117 -.06610 -.376 -.420 -.463 -.504 -.535 -.580 -.644 -.726 -.847 .1179 .117 -.382 -.421 -.464 -.505 -.535 -.580 -.644 -.726 .0238 .026 .117 -.382 -.429 -.466 -.500 -.535 -.580 -.644 -.0887 -.153 .026 .117 -.382 -.429 -.474 -.500 -.535 -.580 -.1436 -.250 -.151 .027 .118 -.382 -.429 -.466 -.505 -.535 -.1965 -.32G -.250 -.151 .027 .118 -.382 -.429 -.464 -.504 -.2114 -.339 -.322 -.249 -.151 .027 .117 -.382 -.421 -.463 -.1733 -.313 -.339 -.322 -.250 -.151 .026 .117 -.382 -.420 -.1372 -.287 -.313 -.339 -.320 -.250 -.153 .026 .117 -.376 -.102


Ace -.509 -.509 -.511 -.518 -.455 -.343 -.234 -.015 .173 -.50710 -.788 -.859 -.932 -1.008 -1.069 -1.160 -1.287 -1.453 -1.694 .1739 .174 -.801 -.860 -.934 -1.011 -1.071 -1.161 -1.288 -1.453 -.0158 -.006 .177 -.798 -.874 -.938 -1.000 -1.071 -1.161 -1.287 -.2347 -.462 -.005 .178 -.796 -.874 -.954 -1.000 -1.071 -1.160 -.3436 -.747 -.464 -.006 .178 -.798 -.874 -.938 -1.011 -1.069 -.4555 -.951 -.745 -.462 -.004 .178 -.796 -.874 -.934 -1.008 -.5184 -1.064 -.951 -.745 -.462 -.006 .178 -.798 -.860 -.932 -.5113 -1.083 -1.064 -.951 -.745 -.464 -.005 .177 -.801 -.859 -.5092 -1.083 -1.083 -1.064 -.951 -.747 -.462 -.006 .174 -.788 -.509

EV Split2 3 4 5 6 7 8 9 10 Ace

reg -.480 -.527 -.578 -.635 -.672 -.638 -.493 -.299 .052 .182dbs -.471 -.518 -.568 -.625 -.662 -.628 -.483 -.290 .052 .182


GLOSSARY

Blackjack has its own vocabulary. This glossarydefines all of the blackjack terminology used in thisbook. Some commonly used words have two meanings inblackjack conversations, and this glossary explains whichof the meanings is used in this book.Ace. An ace is a card that you may count as either one or eleven.

The easiest way to count a hand containing an ace is to countthe ace as eleven initially. If your hand total comes to morethan twenty-one with the ace as eleven, then count the ace asone.

Ace adjustment. If you count aces in addition to your primarycount, when you use your count of aces you are said to bemaking an ace adjustment.

Action. Your action is the total amount you bet on all of yourhands. If you play 100 hands at $25 each, then your action is$2500.

Back count. To back count is to count cards at a table as aspectator, Le., without playing a hand.

Balanced count. This is a counting system in which the positivecards exactly balance the negative cards, so the total countover a complete deck is zero.

GLOSSARY 335

Basic strategy. Basic strategy for blackjack is the set of playingdecisions you should make in order to maximize your advantage if you use no information about cards remaining in thepack except that the cards in your hand and the upcard arenot among them. You are presumed to know the number ofdecks shuffled together, the details of the rules, and whatprocedures the dealer must follow.

Bet variation. In order to get an edge at blackjack, you generallyhave to bet more when you have an edge and less when thecasino has an edge. The ratio of your big bet to your small betis the bet variation.

Black check. Most casinos use black as the color of their $100chips. Black action means bets with $100 chips.

Blackjack. Blackjack is the name of the game. In conversationbut not in this book, blackjack also means a natural.

Bonus. Some casinos pay bonuses for certain hands, such ashands containing six cards or 6-7-8 all in the same suit.

Break the deck. To break the deck is to shuffle earlier than usual.This sometimes is done if a customer is suspected to be a cardcounter. It is also called shuffle up.

Burn. After the shuffle and cut, one card is placed on the bottomof the pack or in the discard tray. This procedure is calledburning a card, and the card temporarily removed from playis called the burned card or bum card.

Bust. To bust is to go over twenty-one. If you bust you lose. If thedealer busts and you do not bust, you win. If both you and thedealer bust, you lose; this is the basis for the casino's edge.

Bustout. bet. This is a side bet offered at some casinos. You canmake it after the dealer's second card is shown, when thedealer has a total of twelve through sixteen. When you makethe bustout bet, you are betting that the next card will be aIO-count card. If it is, your bustout bet wins 2: I.

Camouflage. To use camouflage is to make a bad play in hopes oflooking a gambler rather than a card counter. Camouflagecosts money, so if you do too much of it you give up your edge.

Check. A check or a chip is a form of money in a casino.

Chip. See check.

Counting cards. Card counting means keeping some sort oftrack of which cards have been played. Card counters gener-


ally vary bet size and playing strategy as the composition ofthe remaining pack changes.

Count per deck. Count per deck is the running count divided bythe number of decks or fraction of a deck that you have notseen. Never use this expression in a casino.

Crush count. 9s and lOs count -1; aces and 2s count +2; 3s count+1; and 4s through 8s count zero. This count is used for takingadvantage of the over/under 13 bets.

Cut. When you cut the pack you divide it into two parts. The dealerthen inverts the order of the two parts.

Deal. To deal is to distribute the cards.

Dealer. The dealer is the casino employee who deals the cards.

Deck. This book uses deck to mean 52 cards. A single deck containsfour aces, four each of 2 through 9, and sixteen lOs. Inconversation but not in this book, deck is also used to meanpack.

Discards. Discards are the cards used since the last shuffle. Theyare set aside to be brought into play at the next shuffle.

Double down. When you double down, if your cards are notalready face up you must turn them up. Then double your bet.You will receive one more card. The advantage of doublingdown is that you are allowed to increase your bet after you seeyour hand and the dealer's upcard. The disadvantage is thatyou must take one and only one more card.

Doubling down after splitting. Suppose you split 8-8 against5. If one of your hands catches 3 for eleven, you should like todouble down. At some casinos you are allowed to double downafter splitting a pair, and at others you are not.

Double exposure. The dealer gets two cards face up before youplay your hand.

Early surrender. Early surrender is when you may surrenderbefore the dealer checks for a natural. If the dealer turns overa natural after you have early-surrendered, the early surrender saved you half a bet.

Einstein count. See hi-opt.

Even money. When you have a natural, accepting even money isthe same as putting out an insurance bet. So even money isanother term for insurance, but only if you have a natural.

GLOSSARY 337

Expected win. The average result if you were able to make a betover and over again in the same apparent situation is theexpected win of that bet.

First base. You are at first base if you are playing the spot nearestthe dealer's left hand. The dealer finishes hitting your handbefore moving on to the hands of the other players.

Five-card half win. If you have five cards and you have notbusted, you have the option of turning in your hand andaccepting a win equal to half of your bet. This rule is offeredat some Asian casinos.

Half win. See five-card half win.

Halves count. Aces and lOs count -1; 9s count -0.5; 2s and 7s count+0.5; 3s, 4s, and 6s count +1; 5s count +1.5; and 8s count zero.

Hand. The dealer gives you two cards to start your hand, and youmay take more cards if you wish. All these cards are yourhand. The dealer's cards are the dealer's hand. In conversation but not in this book, hand is also used to mean round.

Hard. Any hand that does not contain an ace is a hard hand. Ahand with an ace can be hard too, if the hand totals twelve ormore and all aces in the hand count one. Examples: 6-7 is hardthirteen; 5-10-ace is hard sixteen. Since twelve is the smallestsoft hand, the word "hard" is not used with hands of eleven orless.

High-low count. Aces and lOs count -1, 2s through 6s count +1,and 7s through 9s count zero.

Hi-opt or hi-opt I. lOs count -1; 3s through 6s count +1; and aces,2s, 7s, 8s, and 9s count zero. Also known as the Einstein count.Sometimes hi-opt I is used to refer to two counts, this one anda side count of aces. In this book, hi-opt is used to mean theprimary count only, with no side count of aces. There also isa hi-opt II, a higher-level count.

Hit. To hit a hand is to take a card. The card you receive when youhit is also called a hit.

Hole card. When the dealer gets a card face up and a card facedown, the hole card is the card that is face down. You are notsupposed to know the value of the hole card until after youplay your hand.

Index number. See strategy number.


Intermediate. An intermediate card is a 7, 8, or 9. lOs and acesare called big cards and 2s, 3s, 4s, 5s, and 6s are called smallcards.

Insurance. Insurance is a side bet that the dealer has a natural.You may buy insurance only when the dealer's upcard is anace. If the dealer has a natural, the insurance bet wins double.If the dealer does not have a natural, the insurance bet loses.

Kelly betting. This means betting a percentage of your bankrollequal to your advantage divided by· the variance of possibleoutcomes. It is also known as proportional betting.

Late surrender. Late surrender is when you must wait tosurrender until after the dealer checks the hole card to see ifthe hand is a natural. If the dealer has a natural you may notsurrender.

Lay and pay. This is one procedure a dealer might use to pick upthe cards. It involves leaving all the cards on the table untilthe payouts have been completed. An alternate procedure ispick and pay.

Lose all to a natural. When you double down or split and thedealer winds up with a natural, you lose both bets. This ruleis common in Europe.

Lose one bet to a natural. When you double down or split andthe dealer winds up with a natural, you lose only your originalbet.

Macao surrender. See five-card half win.

Multiple deck. A multiple-deck game is blackjack using morethan one deck.

Natural. A natural is a two-card hand of twenty-one points. Theonly way to make twenty-one in two cards is ace-IO. Inconversation but not in this book, a natural is sometimescalled a blackjack.

Over/under 13. This is two side bets. Aces count one and thirteenloses. On the over you bet that your first two cards will totalover thirteen. On the under you bet that they will total underthirteen

Pack. This book uses pack to denote any collection of cards, asdistinguished from deck, which is 52 cards.

Pair. A pair is two cards of the same value, such as 7-7 or 8-8.Whether two unlike lOs (such as queen-king) are a pair is up

GLOSSARY 339

to the casino management. Generally a pair may be split andplayed as two hands.

Pat. A pat hand is an unbusted hand worth at least seventeenpoints. A dealer or player who has a pat hand is said to be pat.

Penetration. This word describes how far down into the pack thedealer goes before shuffling. The better the penetration, themore profit potential for a card counter, everything else beingequal.

Pick and pay. This is one procedure a dealer might use to pick upthe cards. It involves picking up the cards on a hand as soonas the bet on that hand has been settled. An alternateprocedure is lay and pay.

Pit. Blackjack tables are not randomly scattered throughout thecasino. Rather, they are arranged in groups called pits.

Pit boss. A pit boss is the casino employee in charge of all thetables in a particular pit.

Proportional betting. Proportional betting is betting a percentage of your bankroll equal to your advantage divided by thevariance of possible outcomes. It is also known as Kellybetting.

Push. A push is a tie, and no money changes hands. A push occurswhen both you and the dealer have unbusted hands with thesame total points. If you bust you cannot push. If ties lose, asthey do in double exposure, there are no pushes.

Random. For the cards in a deck to be random, each card musthave an equal chance of being in each position, and allpossible orderings of the cards must be equally likely.

Rime. A rifile is part of a shuffle. A ritlle is taking two like-sizedgroups of cards and interleaving one into the other.

Resplit. The opportunity to resplit occurs when you split a pairand receive another card of the same value. To resplit meansto put out a third bet and play the cards as three hands. Yousometimes have opportunities for further resplits to makefour or more hands.

Round. One round includes making bets, dealing two cards toeach player and the dealer, serving the players' hands,finishing the dealer's hand, and making the payoffs.


Running count. The running count is the total points yourcounting system assigns to all the cards you have seen sincethe last shuffle. Never use this expression in a casino.

Second-chance 21. This rarely-offered rule allows you to throwin your tirst hand and receive a new hand, at the cost of halfa bet.

Shoe. A shoe is a box for holding cards. Since dealing with morethan two decks held in the hand is awkward, three or moredecks are dealt out of a shoe. Sometimes one or two decks aredealt from a shoe.

Shoe game. A shoe game is a blackjack game in which the dealerdeals out of a shoe.

Shuffle. This book uses shuffle to mean the sum total of all actionsthe dealer uses to reorder the cards. Outside of this book, ashuffle is sometimes used to mean a riffle.

Shuffle up. See break the deck.

Single exposure. This is the traditional variety of blackjack the dealer's hand has one card face up for you to see before youplay your hand.

Snapper. Another word for natural.

Soft. A soft hand contains at least one ace and is valued at twentyone or less when one ace is counted as eleven. Examples: Ace6 is soft seventeen, and ace-6-ace is soft eighteen. Thesmallest soft hand possible is twelve, and the largest istwenty-one.

Split. To split you must have a pair. If your cards are not alreadyface up, tum them up. Then put out an additional bet equalto your original bet, and play the cards as two hands.

Stand. To stand is to take no more cards.

Stiff. A stiff is a hand that is not pat and that may bust if hit once.Stiffs include hard twelve through sixteen. In conversationbut not in this book, stiff has another meaning: a person whodoes not tip.

Strategy number. Comparison of your count per deck to astrategy number tells you whether to hit or stand, or doubledown, etc.

Strip. A strip is part of a shuffle. The dealer holds the deck withone hand and pulls groups of cards off the top with the other

GLOSSARY 341

hand, with the result that the order of the groups in inverted.A strip can also mean a street containing multiple casinos,and when capitalized means Las Vegas Boulevard.

Surrender. To surrender is to give up half of your bet for theprivilege of not playing out your hand. You keep the other halfof your bet.

TARGET. This is an acronym that stands for nonsense. Proponents of TARGET think you can extrapolate past results suchas wins and losses and whether the dealer has been bustingto predict future results.

Team play. This means two or more people playing blackjack inthe same casino and sharing information. It might mean twocard counters at the same table but using different systemsto help each other make better decisions. Or it might be a cardcounter betting small sums who, when the count is high, callsin a partner who makes only big bets.

Third base. You are at third base if you have the spot to thedealer's right. Yours is the last player hand to be finished.

Tie. A tie is when you and the dealer end up with the same totalpoints.

True count. This is another way of saying count per deck.

Twenty-one. This is another name for the game of blackjack.

Unbalanced count. This is a counting system in which the totalpoints in a full deck do not add up to zero.

Upcard. The upcard is the card in the dealer's hand that is faceup for all the players to see before they play their hands.

Win rate. Win rate is how fast you are expected to win. It can beexpressed in percent or in dollars per hour.


SELECTED

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prises, 1981.Uston, Ken. The Big Player. New York: Holt, Rinehart and

Winston, ]977.Uston, Ken. Two Books on Blackjack. Wheaton, MD: Uston

Institute of Blackjack, 1979.Vancura, Olaf & Ken Fuchs. Knock-Out Blackjack. Unstated:

Isochoric Publishing, 1996.Wilson, Allan N. The Casino Gambler s Guide. New York:

Harper & Row, 1966.Wong, Stanford. Basic Blackjack. La Jolla: Pi Vee Press, 1992;

last revised 1993.Wong, Stanford. Betting Cheap Claimers. La Jolla: Pi Vee

Press, 1992.


Wong, Stanford. Blackjack Analyzer (software). La Jolla: PiVee Press, 1992.

Wong, Stanford. Blackjack Count Analyzer (software). LaJolla: Pi Vee Press, 1993.

Wong, Stanford. Blackjack in Asia. La Jolla: Pi Vee Press,1979.

Wong, Stanford. Blackjack Secrets. La Jolla: Pi Vee Press,1993.

Wong, Stanford. Blackjack World. La Jolla: Pi Vee Press,monthly 1979-1984.

Wong, Stanford. Casino Tournament Strategy. La Jolla: Pi VeePress, 1992; last revised 1966.

Wong, Stanford. Current Blackjack News. La Jolla: Pi VeePress, monthly since 1979.

Wong, Stanford. Nevada Blackjack. La Jolla: Pi Vee Press,monthly 1979-1984.

Wong, Stanford. Optimal Strategy for Pai Gow Poker. La Jolla:Pi Vee Press, 1990; last revised 1992.

Wong, Stanford. Professional Video Poker. La Jolla: Pi VeePress, 1988; last revised 1993.

Wong, Stanford. Winning Gamer. La Jolla: Pi Vee Press,monthly 1984-1986.

Wong, Stanford. Winning Without Counting. La Jolla: Pi VeePress, 1978.

Zender, Bill. Card Counting for the Casino Executive. LasVegas: Zender, 1990.


INDEX

Symbols10-count 54-57

Aall-time high 200automatic win 119-124

BBasic Blackjack 23, 24Basic Blackjack Betting 239basic strategy 22-29, 158-

159basic strategy win rate 50Beat the Dealer 31, 149,

219-220, 298benchmark defined 18benchmark win rate 19, 44Blackjack Count Analyzer 4,

34, 36-37, 99, 166, 286287, 292

Blackjack for Blood 213Blackjack Forum 50, 195,

248Blackjack Monthly 145Blackjack Secrets 4, 30,

213, 214

Blackjack World 5bonus for 6-7-8 126-138bonus for 7-7-7 138Braun, Julian 31, 253, 293bustout bet 56-57, 185

CCarlson, Bryce 213Chambliss, Carlson R. 293composition-dependent 23-

24computer 212Cooper, Carl 213count per deck 32-34crush count 145-147Current Blackjack News 5,

34Curtis, Anthony 5

DDalton, Michael 5dealer hits soft 17 44, 48,

182, 185double down 25, 40, 58double exposure 156-182

INDEX

Douglas, Dave 5, 224-231Dubner, Harvey 31

Eearly half-win 105early surrender 90Einstein, Charles 239European rules 81-85expected value 187-188expected win 183-189

Ffive-card half win 99-105,

185French, Marvin 200

GGaem, Bill 223generic basic strategy 24-29Giles, Peter 199, 224Griffin, Peter 23, 31, 34,

53, 150, 200, 214, 250,301

Gwynn, John M. Jr. 201,246

Hhalves count 148-151, 174-

175, 185, 215, 270hard hand 28, 43hi-opt 174, 185, 213-215high count 216How to Play Winning Black

jack 31, 293Humble, Lance 213

incomplete shuffle 223insurance 29, 37, 52-57,

185

Jjumping into a game 36

347

KKelly criterion 203-205

Llate surrender 92lose all to a natural 81-85,

185

Mmultiple-card surrender 94

95

Nnadir 195-199Nevada Blackjack 5no dealer natural 85-88no hole card 80-88, 90, 92normal curve 192-200

oOlsen, Eddie 239optimal bet size 201-206optimal number of simulta-

neous hands 208-211over/under 13 16, 142-147,

185"Over/Under Report,

The," 145overbetting 206-207

p

Patterson, Jerry 239player edge 35, 157Playing Blackjack as a

Business 213, 293Playing Blackjack in Atlantic

City 293Polo, Frank 5proportional betting 20 I

Rresolving toss-ups 218resplits 68, 182, 185


Revere, Lawrence 31, 213,293

Review Journal 199risk 186-187, 189-200Roginski, Thomas C. 293running count 32

Ssampling error 19-20Schlesinger, Donald 5, 50,

199-200second-chance 21 96-97Seri, Armand 201seven cards 98-124shuffle 222-251shuffle tracking 212side counts 217six cards 98-124, 185Smallwood, Jake 145Snyder, Arnold 53, 145-

146, 246soft hand 25-28, 42Speer, John 5split 24-25, 38-39, 67-69standard casino shuffle 224-

231standard deviation 190-192Stanford Wong's Blackjack

JVelVsletter 5stopping rules 207-208streakiness 239-251surrender 28, 89-97, 185surrender after doubling

down 96

TTARGET 239Theory of Blackjack, The 23,

31, 34, 150, 301Thorp, Edward O. 31, 149

150, 219-220, 298-300total-dependent 23-24two-card combinations 217

218

uUston, Ken 199

Wwin rates 17-21, 185win rates, double expo-

sure 182World's Greatest Blackjack

Book, The 2I3

349

PUBLICATIONS BY

STANFORD WONG

BooksProfessional Blackjack. This 350-page book has

a complete and accurate presentation of the high-low,the counting system used by more card counters thanany other because of its combination of simplicity andpower. If you want a more advanced counting system,Professional Blackjack also contains the halves.

Blackjack Secrets. This 250-page book explainshow to get away with playing a winning game of blackjack in casinos. (You can't win money just by playingyour cards well; you also must avoid drawing attentionto your card-counting ability.) It also contains an introduction to the high-low card counting system.

Basic Blackjack. This 250-page book is a comprehensive presentation of basic strategy and win ratesfor all common rules and most exotic rules for the game


of blackjack. It also explains methods of getting anedge that do not involve counting cards.

Casino Tournament Strategy. This is the bookthat explains how to get an edge over the other playersin tournaments. In 350 pages it covers blackjack, craps,baccarat, keno, and horses. The strategies required forsuccess in a blackjack tournament are completely different from the strategies that are appropriate forblackjack away from tournaments.

Professional Video Poker. This is the book thatshows how to get an edge on video poker at bothNevada and Atlantic City casinos. The material wasdeveloped with the aid of two video-poker pros. (Yes,there are people who support themselves playing videopoker.)

Optimal Strategy For Pai Gow Poker. This isthe book that shows how to get an edge at pai gowpoker.. Mike Caro, author of Professional Pai Gow PokerReport, says "Those serious players seeking to make aliving at pai gow poker will find Wong's Optimal Strategy for Pai Gow Poker an almost perfect path to profit."

NewsletterCurrent Blackjack News is a monthly newslet

ter for blackjack players. It is available electronically,and the' entire content is posted on the bj21.com Internet site. Current Blackjack News is also available byregular mail.

When a profitable casino opportunity arises between regular monthly issues, a special issue is published. Special issues have been about such things ascasinos paying 2:1 on naturals or offering early surren-

351

der, blackjack with jokers added to the shoe, and doubleexposure blackjack with 3:2 on naturals.

Internet Web SiteOn bj21.com you will find stimulating discus

sions on winning money from casinos, mostly by playing blackjack. Read messages posted by other blackjack players, post your own message if you wish, andchat. There are both free pages and pages requiringpayment of a meml>ership fee.

SoftwareBlackjack Count Analyzer. This is a computer

program for playing blackjack. It works on IBM PCsand compatible conlputers. Besides playing blackjack,Blackjack Count Afzalyzer generates tables of strategyindexes and runs lligh-speed simulations. It handlesall common and uncommon rules variations, and alsosome rules variations that have never been offered inany casino.

A simplified version of it, Blackjack Analyzer, alsois available; Blackjack Analyzer uses basic strategyonly, whereas Blackjack Count Analyzer supports cardcounting.

Tournament Blackjack. This is a compllter program for playing blackjack tournaments. It works onIBM PCs and compatible computers. It can be used toanalyze tournament situations by simulation.

Stanford Wong Video Poker. This program, likeits predecessor Video Poker Analyzer, is a computerprogram for playing and analyzing video poker. It workson IBM pes and compatible computers.


Also available is VPEXACT, a program whoseonly function is to cycle through all possible hands ofvideo poker to calculate the payback to the customer.

Ordering InformationWrite to Pi Vee Press, 7910 Ivanhoe Ave #34, La

Jolla, CA 92037-4511 for prices and an order form. Oremail [email protected]. Or call (619) 456-4080. Orfax (619) 456-8076.

professional blackjack

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