-The Mathematics of Poker BillChen Jerrod Ankenman Other ConJelCo titles: Cooke's Rules of Real Poker by Roy Cooke and JohnBond Hold'em Excellence by Lou Krieger How to Play Like aPoker ProbyRoy Cooke and John Bond How toThink Like a Poker ProbyRoy Cooke and John Bond Internet Poker by Lou Krieger and Kathleen Watterson Mastering No-Limit Hold'em by Russ Fox and Scott T.Harker MoreHold' em Excellence by Lou Krieger Real Poker II:ThePlay of Hands 1992-1999 (2"ded.) by Roy Cooke and John Bond Serious Poker by Dan Kimberg SteppingUpby Randy Burgess TheHomePoker Handbook by Roy Cooke and John Bond WhyYouLose at Poker by Russ Fox and Scott T.Harker WinningLow-Limit Hold 'emby Lee Jones WinningOmahal8 Poker by Mark Tenner and Lou Krieger WinningStrategies for No-Limit Hold' em by Nick Christenson and Russ Fox VideoPoker - Optimum Play by Dan Paymar Software StatKing for Windows The Mathematics of Poker BillChen Jerrod Ankenman ConJelCo LLC Toolsforthe Intelligent Gambler tm Pittsburgh, Pennsylvania ( The Mathematics of Poker Copyright2006 by BillChenandJerrod Ankenman Allrightsreserved. This book may not be duplicated in any way or stored in aninfonnation retrievalsystem,withouttheexpresswrittenconsentof thepublisher,exceptintheform of brief excerptsorquotationsforthepurposeof review.Makingcopiesof thisbook,or anyportion,foranypurposeotherthanyourown,isaviolationofUnitedStates copyright laws. Publisher's Cataloging-in-Publication Data Chen, Bill Ankenman, j errod The Mathematicsof Poker x,382p.; 29cm. ISBN-13:978-1-886070-257 I SBN-I0:H86070-25-3 I. Ticle. Library of Congress Control Number: 2006924665 Cover design by Cat Zaccardi Book design by Daniel Pipitone Digital production by Melissa Neely ConJelCo LLC 1460 Bennington Ave Pittsburgh, PA15217 [412] 621-6040 http://www.conje1co.com First Edition 57986 Errata, if any.can be foundat http://www.conjelco.comlmathofpoker/ able of Contents J.dGlowledgments ...V11 .. . ... . . . . . ........ IX uction .. .... .. ... 1 pter 1Decisions Under Risk:Probability and Expectation.. .... 13 pt er 2Predicting the Future:Variance and Sample Outcomes.. .............. 22 3Using All the Information:Estimating Parameters and Bayes' Theorem .. 32 ?:i:rt II : Exploitive Play pter 4Playing the Odds: Pot Odds and Implied Odds .. pt er 5Scientilic Tarot: Reading Hands and Strategies.. ... ... .. . . Olapter 6The Tellsare in the Data: Topicsin Online Poker .. . . ..... . . apter 7Playing Accurately, Part I:Cards Exposed Situations apter 8Playing Accurately, Part II:Hand vs.Distribution .. apter 9Adaptive Play:Distribution vs.Distribution. ;>-..rtII I: Optimal Play apter 10Facing The Nemesis: Game Theory .. .... . . . . . . apter 11One Side of the Street:Half-Street Games.. .. . . ....... . . . .... . Chapter 12Headsup With High Blinds: ThcJam-or-Fold Game...... . . .. . . . . .. 47 . . .. . 59 . .... 70 ... . . 74 . ... 85 94 . .lOl . ... . . 111 Chapter 13Poker Made Simple: The AKQGame .... . .... . ... . ...... . .. . ..... . . ..... 123 ... . . . 140 Chapter 14YouDon't Have ToGuess: No-LimitBet Sizing. Chapter 15Player XStrikes Back: Full-Street Games. Appendix to Chapter 15 The No-Limit AKQGame. Chapter 16Small Bets, Big Pots:No-Fold[O,IJGames .... .. .... . . Appendix to Chapter 16 Solving the Difference Equations. Chapter 17Mixing inBluITs:Fmite Pot [O,IJ Games.. . ..... . ..... . . . . . _. Chapter 18LessonsandValues: The [O,1J Game Redux. Chapter 19The Road to Poker:Static Multi-StreetGames. . Chapter 20Drawing Out: Non-Static Multi-Street Games.. ..... . . . Chapter 21A Case Study: Using Game Theory.. .... .. ...... .. _... . Part IV: Risk _. .. 148 . ..... 158 . . . ... . 171 _.. _. 178 . . ..... . . . . . . 195 198 . . . 216 . .. . . ... 234 ..249 . ... 265 Chapter 22Staying in Action:Risk of Ruin.. ... . . 281 Chapter 23Adding Uncertainty: Risk of Ruin with Uncertain Wm Rates... ... 295 Chapter 24Growing Bankrolls: The Kelly Criterion and Rational Game Selection ... 304 Chapter 25Poker Fmance: Portfolio Theory and Backing Agreements .. .310 Part V:Other TopiCS Chapter 26Doubling Up: Tournaments, Part I. .. . ... . ...... . Chapter 27Chips Aren't Cash:Tournaments, Part II .. Chapter 28Poker's StillPoker: Tournaments, Part III . Chapter 29TIrree' s a Crowd: Multiplayer Games .. Chapter 30Putting It All Together:Us;ng Mathto Improve Play .. Recommended Reading. About the Authors .. . About the Publisher .. . THE MATHEMATICS OF POKER . .. 321 . .... . . . . 333 . ..... . . 347 . ... 359 . . ..... 370 . ... . 376 . . .... . . . . 381 . ... 382 v vi THEMATHEMATICS OF POKER Acknowledgments A book like thisisrarely the work of simply theauthors; lllany people have assistedus along the way, both in understanding poker and in the specific task of writing down many of the ideas thatwehavedevelopedoverthelastfewyears.AbookcalledTheMatlu:maticrtf IM.erwas conceived by Bill, Chuck Weinstock,andAndrew Latta several yearsago, beforeJerrodand Bill had even met. TIlat book was set to be a fairly fOlmal,textbook-like approach to discussing the mathematical aspects of the game. A few years later,Jerrod and Bill had begun [0 collaborate on solving some poker games and the idea resurfaced as a book that was less like a mathematics paper and more accessibletoreaders with a modest mathematical background. Our deepest thanks COthose who read the manuscript and provided valuable feedback. Most notable among those wereAndrew Bloch,Andl'CWProck, and AndrewLacco,who scoured sectionsindetail,providingcriticismsandsuggestionsforimprovement.AndrewProck's PokerStove [001(lutp:/lww\v.pokerstove.com) wasquite valuable in performing many of the equity calculations. Others who read the manuscript and provided useful feedback were Paul R. Pudaite and Michael Maurer. JcffYass at SusquehaJma International Group (http://www.sig.com)has been such agenerous employer in allowing Bill to work on poker, play the World Series, and so on. We thank Doug Costa, Dan Loeb,Jay Siplestien, andAlexei Dvoretskii at SIG fortheir helpful comments. We have learned a great deal from a myriad of conversations with various people in the poker community.OUffriend and partner Matt Hawrilenko, andformerWSOP champion Chris Ferguson aretwoindividualsespeciallyworth si.ngling out fortheirinsight and knowledge. Both of us participate enthusiastically in the *J\RG community, and discussions with members of thatcommunity,too,havebeenenlightening,particularlyroundtablediscussionswith players including Sabyl Cohen,J ackMahalingam,JP Massar, Patti Beadles,SteveLandrum, and (1999 Tournament of Champions winner) SpencerSun. Sarah Jennings, our editor, improved thebook significantlyin ashort period of time bybeing willing to slog drrough equations and complain about the altogether too frequent skipped steps. Chuck WeinstockatConjelcohasbeenamazinglypatientwithallthedelaysandextended deadlines that corne from working with authorsfor whom writing isasecondary occupation. We alsoappreciatethecommentsandencouragementfromBill'sfatherDr.An-BanChen whohaswrittenhisownbookandhasnumerouspublicationsinhisfieldofsolidstate physics. Jerrod' smother Judy hasbeen aconstant source of support forall endeavors,evcn crazy-sounding oneslikeplaying poker for a living. TIrroughout the writing of thisbook, as in the rest of life, Michelle Lancaster hasconstantly been wonderful and supportive ofJ errod. This book could not have been completedwithout her.Patricia Walters has also provided Billwith support and ample encouragement, and itis of no coincidence that the bookwascompleted during the three years she hasknown Bill. THE MATHEMATICSOF POKERvii viiiTHE MATHEMATICS OF POKER Foreword ::lex:': believeaword I say. !: ~::lO(thatI'm lying when Itellyouthatthisisanimportant book.Idon' tevenlieatthe :ci-c table - not much,anyway -- so why would I lieabom a book Ididn't even write? ::"3just that you can't trust metobe objective.Iliked this book before I'd even seen a single .?4.....I li."-ed it when it was just a series of cOllversations betvveen Bill,myself, and a handful ::i other math geeks.Andif Ihadn' tmadeupmy mindbeforeI'dreadit,I'm pretty sure ~ - ' dhave \von me over with thefirstsentence. :Jon"tworry,though.Youdon' thavetocrustme.Mathdoesn'tlie.Andresultsdon'tlie, -neroIn the 2006 WSOP, the authorsfinishedin the money seven times,includingJerrod's ~ n dplacefinishin LimitHoldern,andBill'stwowinsin Limit andShortHandedNo :....:rnit Holdcm. -'lost poker booksgetpeopletalking.The best booksmakesomepeoplesay,"Howcould nyone publishour carefullyguardedsecrets?"Othertimes,youseestuff thatlooksfishy Cloughtomake you wonder if the author wasn't deliberately giving out bad advice.Ithink :hisbook will get people talking, too, but it won't bethe usual sort of speculation.No oneis going to argue that BillandJerroddon't know their math. The argument will be about whether or not themath is important. People like to talk about poker as"any man's game." Accountams and lav"yers, students and housewives can all compete at the same level-- all you need isa buy-in,some basic math and goodintuitionandyou,too,cangettothefinaltableof theWorldSeriesofPoker. That norian isespecially appealing tolazy people who don't wanetohave tospend yearsworking atsomething toachievesuccess.It'strueinthemost literalsensethatanyonecanwin,but with some well-invested effort, you cantipthe scalesconsiderably in your favor. The math in here isn't easy_You don' t need a PhD in game theory to understand the concepts in thisbook, but it's not assimple as memorizing starting hands or calculating the likelihood of makingyour Hushontheriver.There'ssomeworkinvolved.The peoplewhowantto believe inruitionisenough aren' t going to readthis book.But the people who make the effort willbe playing with adefiniteedge.Infact, much of my poker successistheresult of using some of the most basic concepts addressed in thisbook. Bill andJerrod have saved you a lot of time. They've saved me a lac of a time, too.I get asked alotof pokerquestions,andmostareprettyeasytoanswer.ButPveneverhadagood response when someone asksmeto reconunend abook for understanding gametheory asit relatesto poker. I usually end up explaining that there are good poker books and good game theory books, but no book addressesthe relationship between the two. Now Ihave an answer.And if Iever findmyself teach.ingapoker classforthemathematics department at UCLA, thiswill be the only book on the syllabus. Chris 'Jesus" Ferguson Champion, 2000 World Seriesof Poker ::\ovember 2006 THE MATHEMATICS OF POKERix xTHEMATHEMATICS OF POKER Introduction (7fyouthinkthe math isn't important) youdon't know the right math." Chris "Jesus" Ferguson, 2000 World Series of Poker champion Introduction Introduction Inthe late1970sandearly1980s,thebond andoption marketsweredominatedbytraders who had learnedtheir craft by experience. They believed that their experience and inruition fortrading were arenev..ableedge;thatis,thatthey couldmakemoney just astheyalways had by continuing to trade asthey always had. Bythe mid-1990s, a revolution in trading had occurred;theoldschoolgrizzledtradershadbeen replacedby ancw breedof quantitative analysts,applying mathematics tothe "art" of trading and making of it a science. If thelatestbackgammonprograms,basedonneuralnettechnologyandmathematical analysishad played in a tournament in the late1970s,their play would havebeen mocked as overaggressive and weak bythe experts of the time.Today,computer analyses are considered to be the final word on backgammon play by the world's strongest players - andthe game is fundamemally changed forit. Andfordecades,thehighestlevelsof pokerhavebeendominatedbyplayerswhohave learnedthegame by playing it,"road gamblers"who have cultivatedintuitionforthegame and are adept at reading other players'handsfrombetting patternsand physicaltells. Over thelastfivetotenyears,awholenewbreedof playerhasrisentoprominencewithinthe pokercommunity.Applyingthetoolsof computerscienceandmathematicstopokerand sharing information across the Internet, these players have challenged many of the assumptions that underlierraditionalapproachesto thegame.One of the most important featuresof this newapproachtothegameisarelianceonquantitativeanalysisandtheapplicationof mathematics tothe game. Our intent in this book isto provide an introduction toquantitative techniques as applied to poker and to the application of game theory, a branch of mathematics, to poker. Any player who plays poker isusing some model, no matter what methods he usestoinform it.Even if a player isnot consciously using mathematics, a model of the siruation isimplicit in his decisions; that is, when he calls, raises, or folds, he ismaking a statement about the relative values of those actions. By preferring one action over another, he articulates his belief that one actionisbettcrthananotherinaparticularsituation.Mathematicsareaappropriate tool for making decisions based on information.Rejecting mathematicsasatool for playing poker puts one's decisionmaking atthe mercy of guesswork. Common Misconceptions Wefrequentlyencounterplayerswhodismissamathematicalapproachout of hand,often basedontheirmisconceptionsabout whatthisapproachisallabout.vVelistafewof these here;theseareideasthatwehave heard spoken,evenbyfairlyknowledgeableplayers.For eachof these,weprovideabrief rebuttalhere;throughoutthisbook,wevvillattemptto present additionalrefutationthrough our analysis. 1)By analyzingwhat has happened inthepast- our opponents, their tendencies, and so on- we canobtain a permanent andrecurring edge. TIlls misconception isinsidious because it seemsvery reasonable; in fact,we can gain an edge overour opponentsbyknowingtheirstrategiesandexploiting them.But thisedgecanbe onlytemporary;ouropponents,evensomeof theoneswethinkplaypoorly,adaptand evolve by reducing the quantity and magnitude of clear errorsthey make and by attempting to counter-exploit us. We have christened thisfirstmisconception the "PlayStation theory of THE MATHEMATICS OF POKER 3 Introduction poker" - that the poker world is full of players who play the same fixed strategy, and the goal of playing poker isto simply maximize profit against the fixed srrategies of our opponents. In fact,ouropponents'strategiesaredynamic,andsowemustbedynamic;noedgethatwe have isnecessarily permanent. 2)Mathematical play is predictable and lacks creativity. In some sensethis istrue;that is,if a player were to play the optimal strategy to agame,his srrategy wouldbe "predictable"- but there wouldbenolhing at allthat couldbedone with thisinformation. In the latter parts of the book, we vvill introduce the concept of balance - this is the idea that each action sequence contains a mixture of hands that preventsthe opponent fromexploitingthestrategy.Optimalplayincorporatesapreciselycalibratedmixtureof bluffs,scmi-bluffs,andvaluebetsthatmakeitappear entirelyunpredictable."Predictable" connotes"exploitable,"butthisisnotnecessarilytrue.If aplayerhasaceseverytimehe raises,thisispredictableand exploitable.However,if aplayer alwaysraiseswhenheholds aces,thisisnotnecessarilyexploitableaslong ashealso raiseswith some other hands. The opponent isnot able to exploit sequencesthat contain other actions because it is unknown if the player holds aces. 3)Math is not always applicable; sometimes lithe numbers go out the window!' This misconception is related to the idea that for any situation, there is only one mathematically correctplay;playersassumethatevenpla)'ingexploitively,thereisacorrectmathematical play- but thattheyhavea"read"whichcausesthemtopreferadilferent play.Butthisis simply anarrow definition of "mathematicalplay" - incorporating new infonnation into our wlderstandingof our opponent'sdistributionandutilizingthatinfonnationtoplaymore accuralely is the major subject of Part II.In fact, mathematicscontains tools(notably Bayes' theorem) that allow us to precisely quantify the degree to which new information impacts our thinking; in fact, playing mathematically ismore accurate asfar asincorporating "reads" than playing by "feel." 4)Optimal play is an intractable problem for real-life poker games; hence, we should simply play expJoitively. TIllsisanimportantidea.Itistrueiliatwecurrentlylackthecomputingpowertosolve headsupholdemorothergamesof similarcomplexity.(\tVcvvilldiscusswhatitmeansto "solvt:"a game in Part III).We have methods that are known to findthe answer,but they will notrun on modem .computersinanyreasonableamount of time."Optimal"play doesnot even exist for multiplayer games, as we shall see. But thisdoes not prevent us from doing two things : attemptingtocreatestrategieswhichsharemanyof thesamepropertiesasoptimal strategiesandtherebyplayina"near-optimal"fashion;andalsotoevaluatecandidate strategiesand findom how far away fromoptimal they are by maximally exploiting them. 5)When playing [online, in a tournament, in high limit games, in low limit games..], youhave to change your strategy compl etely to win. This misconception is part of a broader misunderstanding of the idea of a "strategy" - it is in facttruethatinsomeofthesesituations,youmusttakedifferentactions,particularly exploitively,inordertohavesuccess.But thisisnotbecausethegamesarefundamentally different;it isbecausetheother playersplay differentl y and soyourresponsestotheir play take different forms.Consider for a moment a simple example. Suppose you are dealt A9s on the button in a fullring holdem game. In a small-stakes limit holdem game, six players might limptoyou,andyoushouldraise.Inahighlimitgame,itmightberaisedfrommiddle position,andyouwouldfold.In atournament,itmightbefoldedtoyou,andyouwould raise.These are entirelydifferent actions, bur the broader strategy isthe same in all - choose the most profitable action. 4THEMATHEMATICS OF POKER Introduction lbroughout thisbook, wewill discussa wide variety of poker topics,but overall,our ideas couldbe distilled toone simple piece of play advice:Maximiu average profit. TIllsidea isat the heart of all our strategies, and this isthe one thing that doesn't change from game condition to game condition. Psychological Aspects Poker authors, when facedwith a difficult question, are fond of falling back on theold standby, .lIt depends." - on the opponents, on one's 'read', and so on. And it is surelytrue that the most profitableaction in many poker siruations does in factdepend on one's sense, whether intuitive or mathematical, of what the opponent holds(or what he can hold). But one thing that isofcen missingfromthequalitativereasoningthalaccompanies"Itdepends,"isarealansweror a methodologyforanivingatanaction.Inreality,theanswerdoesinfactdependonour assumptions, and the tendencies and tells of our opponents are certainly something about which reasonable people can disagree. But once we have characterized their play into assumptions, the methods of mathematicstake over and intuition failsasa guide to proper play. Some may take our assertion that quantitative reasoning surpasses intuition as a guide to play asa claim that the psychologicalaspects of poker are without value.But we do not holdthis view.The psychology of poker can be an absolutely invaluabletoolforexploitive play, and the assumptionsthat drive the answersthar our mathematical models can generate are often strongly psychologicalin narure.The methodsby which we utilizethe infonnation that our intuitionor people-reading skillsgiveusisour concernhere. In addition,wedevotetimeto the question of what we ought to do when we are unable to obtain such infonnation, and also inexposing someof thepoor assumptionsthatoftenunderminetheinfonnation-gathering effortsof intuition.'With that said, wewill generally,excepting a fewspecific sections,ignore physicaltellsandopponentprofilingasbeingbeyondthescopeofthisbookandmore adequately covered by other -writers,particularly in the work of Mike Caro. About This Book Wearepracticalpeople- wegenerallydonotstudypokerfortheintellecrualchallenge, although it rums out that there is a substantial amount of complexity and interest to the game. We study poker:with mathematicsbecausebydoing so, wemakemoremoney.Asa result, we are very focused on the practical application of our work, rather than on generating proofs or covering esoteric,improbable cases. TIlls isnot amathematicstextbook, but a primer on theapplicationof mathematicaltechniquestopoker andin howtotum theinsightsgained into increased profit at the table. Certainly, there are mathematical techniques that can be applied to poker that are difficult and complex. But we believe that most of the mathematics of poker isreally not terribly difficult, and we have sought to make some topics that may seem difficult accessible to players without avery strong mathematical background.Btl[on the other hand, itis math, and wefearthat if you areafraidof equationsand mathematical tenninology,it will be somewhat difficultto follow some sections.But the vast majority of the book should be understandable to anyone who hascompletedhighschoolalgebra.Wewill occasionally referto resultsor conclusions from more advanced math. In these cases, it isnot of prime importance that you understand exacdy the mathematical technique that was employed. The important element isthe concept - it isvery reasonableto just "take our word for it" in some cases. THE MATHEMATICS OF POKER 5 Introduction Tohelpfacilitatethis,wehavemarkedoff thestartandendof someportionsr:i.. thetextsothatour lessmathematicalreaderscan skipmore complexderivations.. Just look forthisicon for guidance, indicating these cases.~ In addition, Solution: Solutionsto example problems are shown in shaded boxes. As we said, this book isnot a mathematical textbook or a mathematical paper to be submitted toa journal.The materialhereisnot presented inthemarmer of formalproof,nor dowe intend it to be taken as such. 'Ve justify our conclusions with mathematical arguments where necessary and with intuitivesupplemental argumentswherepossiblein ordertoattemptto maketheprinciplesofthemathematicsofpokeraccessibletoreaderswithoutaformal mathematical background,and wetry not tobe boring. 1ne primary goal of our work here isnot to solve game theory problems for the pure joy of doing so; it istoenhance our ability towin money at poker. This book isaimedat awiderangeof players,fromplayerswith only amodest amount of experienceto world-classplayers. If you have never playedpoker before,thebestcourseof actionistoput thisbook dovvn,read some of theother books inprint aimedatbeginners. play some poker, learn some more, and then return after gaining additional experience. If you are a computer scientist or options trader who has recendy taken up the game, then welcome. This book isfor you. If you are one of a growing classof players who has read a few books, played for some time, and believe you are a solid, winning player, are interested in making the nextstepsbutfeelliketheexistingliteratmelacksinsightthatwillhelpyoutoraiseyour game, then welcome. This book isalso for you. If you are the holder of multiple World Series of Poker bracelets who plays regularly in the big game at the Bellagio,youtoo are welcome. There islikely a fairamount of material here that can help you aswell. Organization The book isorgariized asfollows: Part I: Basics, isan introduction toanumberof general conceptsthat apply toallformsof gamblingandothersituationsthatincludedecisionmakingunderrisk.Webeginby introducingprobability,acoreconceptthatunderliesallof poker.Wethenintroducethe concept of aprobabilitydistribution,an important abstractionthatallowsustoeffectively analyze situationswith a large number of possibleoutcomes,each with unique and variable probabilities. Once we have a probability distribution, we can define expected value, which is the metric that we seek to maximize in poker. Additionally, we introduce a number of concepts fromstatisticsthathavespecific,common,andusefulapplicationsinthefieldof poker, including one of the most powerful concepts in statistics,Bayes'theorem. 6THE MATHEMATICS OF POKER Introduction PartII:ExploitivePlay,isthebeginningof ouranalysisofpoker.Weintroducethe concept of a toy game, which is a smaller, simpler game chat we can solve in order to gain insight about analogous,morecomplicated games.Wethenconsiderexamplesof toy gamesinanumber of situations. First we look at playing poker withme cards exposed andfindthattheplayin manysituationsisquiteobvious;atthesametime,wefind interestingsituationswithsomecounter-intuitivepropertiesthatarehelpfulin understandingfullgames.Thenwe considerwhat manyauthorstreatastheheart of poker,the situation where we play our single hand against a distribution of the opponent's hands and attempt to exploit his strategy, or maximize our win against his play.This is the subject of the overwhehning majority of the poker literature.But we go further, to the in our view)muchmore important case,where wearenot only playing asinglehand against the opponent, but playing an entiredistribution of hands against his distribution of hands.It isthisview of poker,we claim, that leadsto truly strong play. Part III: Optimal Play, isthe largest andmost important part of thisbook.Inthispart, weintroduce the branch of mathematics calledgametheory. Game theory allowsusto findoptimal strategiesfor simple gamesand to infer characteristicsof optimal strategies for more complicated games even if we cannot solve them direcdy. We do work on many variationsof theAKQ,game,asimpletoygameoriginallyintroducedtousin Card Player magazine by Mike Caro. Wethen spend a substantial amount of time introducing and solving [0,1] poker games, of the type introduced by John von Neumann and Oskar in theirseminaltext on game theoryTheory0/ Gamesand Economic Behavior 1944), but with substantially more complexity and relevance toreal-life poker.We also e:-..-plain and provide the optimalplay solution to short-stack headsup no-limit holdem. PartIV:BankrollandRiskincludesmaterialof intereston averyimportanttopicto anyone whoapproachespoker seriously.Wepresenttheriskof ruinmodel,amethod forestimatingthechanceof losingafixedamountplayinginagamewithpositivc expectation but some variance. We then extendthe risk of ruin modelin a novelwayto include the uncertainty surrounding any observation of win rate. We also addresstopics suchasthe Kellycriterion,choosing an appropriategame level,andtheapplicationof portfolio theory to the poker metagame. Part V: Other Topicsincludesmaterial on other important topics.'lburnamenrs are the fastest-growingandmostvisibleformof pokertoday;weprovidcanexplanationof conceptsandmodelsforcalculatingequityandmakingaccuratedecisionsinthe tournamentenvirorunent.WeconsiderthegametheoryofmuItiplayergames,an importantand verycomplexbranch of gametheory,and show somereasonswhyme analysisof suchgamesissodifficult.Inthissectionwealsoarticulateand explain our strategic philosophy of play,including our attemptstoplay optimally or at least pseudo-optimallyaswellasthe situations in which we play exploitively. -:HE MATHEMATICS OF POKER7 Introduction How This Book Is Different This book differsfromother poker books in a number of ways.One of the most prominent isin itsemphasis on quantitative methodsand modeling.Webelieve that intuition isoften a valuabletoolforunderstanding what ishappening.But at the same time,weeschewitsuse asaguidetowhatactiontotake.WealsolookforwaystoidentifysituationswhereOur intuitionisoften wrong, and attempt toretrain it in such situationsin order toimprovethe quality of our reads and our overall play. For example,psychologists have identified that the human brain isquite poor at estimating probabilities,especially for situationsthat occur with lowfrequency. Bycreating alternatemethodsforestimating these probabilities, we can gain an advantage over our opponents. It is reasonable to look at each poker decision asa two'part processof gathering information and then synthesizing that information and choosing the right action. It is our contention that inruition has no place in the latter. Once we have a set of assumptions about the situation - how our opponent plays,what our cardsare,the pot size,etc.,thenfindingtherightactionisa simplematter of calculating expectation forthe various optionsand choosing the option that maximizesthis. The second major way in whichthisbook differsfromother poker books isin itsemphasis on strategy,contrastedto an emphasis on decisions.Many poker books divide the hand into sections, such ascC preflop play,""flop play,""turn play,"etc. By doing this, however,they make itdifficulttocapturethewayin whichaplayer'spreflop,Hop,rum,andriverplayareall intimatelyconnected,and ultimately part of the same strategy.Wetry tolookat handsand gamesinamuchmoreorganicfashion,where,asmuchaspossible,theevaluationof expectation occursnot at eachdecisionpoint but at the begirming of the hand,whereafull strategy forthe gameischosen.Unfortunately, holdem and other popular poker gamesare extraordinarilycomplexinthissense,andsowemustsacrificethissometimesdueto computationalinfeasibility.Buttheideaof carryingastrategyfonvardthroughdifferent betting rounds and being constantly aware of the potential hands we could hold atthis point, which OUTfellowpoker theorists Chris Ferguson and Paul R.Pudaite call "reading your own hand," isessential toour view of poker. Athird way inwhichthisbook differs frommuch of the existing literature isthat itisnot a book about howto playpoker. It isabook abouthowtothink about poker. We offer very littleintermsof specificrecommendationsabout howtoplayvariousgames;insteadthis book isdevotedtoexaminingtheissuesthatareof importanceindeterminingastrategy. Insteadof aroadmaptohowtoplaypokeroptimally, we insteadtry toofferaroadmapto how tothink about optimal poker. Our approach to studying poker,too, diverges from much of the existing literarure. We often work on toygames, small,solvablegamesfromwhichwehopetogaininsightintolarger, more complex games. In a sense, we look attoy gamesto examine dimensions of poker, and howthey affect our strategy. How doesthe game change when we move fromcards exposed to cardsconcealed? From gameswhere playerscannot foldto games where they can? From games where the first player always checks to games where both players can bet? From games with one street to games with cwo? We examine these situations by way of toy games - because toy games, unlike realpoker, are solvable in practice - and attempt to gaininsight into how we should approach the larger game. 8THE MATHEMATICS OF POKER Introduction Our Goals It isour hope that our presentation of thismaterial will provide at least twothings; that it will aid youto play more strongly in your own poker endeavorsandtothink about siruanons in poker in a new light, and that it will serve as a jumping-off point toward the incredible amount of seriouswork that remainstobedone in thisfield.Poker isin a critical stage of growthat thiswriting;the universeof pokerplayersandthemainstream credibilityof thegamehave never been larger.Yetitisstilllargely believed that intuition and experience aredetennining factorsof thequalityof play- just asinthebondand optionsmarketsintheeady 19805, trading was dominated by old-time veterans who had both qualitiesin abundance.A decade later, the quantitative analysts had grasped control of the market, and style and intuition were onthedecline.In thesameway,eventhosepokerplayersregardedasthestrongestinthe worldmake serious errors and deviationsfromoptimal strategies.1bis isnot an indictment of their play, but a reminder that the distance between the play of the best players in the world andthebestplaypossibleisstilllarge, andthatthereforethereisalargeamount of profit available tothose who can bridge that gap. THE MATHEMATICS OF POKER9 Introduction 10THE MATHEMATICS OF POKER -Part I: Basics ((As for asthelaws qf mathematics rqer toreality,theyare not certain; as for astheyarecertain,theydo not rifer toreality." Albert Einstein Chapter 1-DecisionsUnder Risk:Probabil ity andExpectation Chapter 1 DecisionsUnder Risk:Probability andExpectation There are as many different reasons forplaying poker as there are players who play the game. Someplayforsocialreasons,tofeelpartof agroupor"oneof theguys,"someplayfor recreation, just to enjoy themselves.Many play forthe enjoyment of competition. Still others playtosatisfygamblingaddictionsortocoverupotherpainintheirlives.Oneofthe di.fficultiesof taking amathematical approachto these reasons isthat it'sdifficultto quantify the valueof having fun or of a sense of belonging. In additionto some of the more nebulous and difficult to quantify reasons forplaying poker, theremayalsobeadditionalfinancialincentivesnotcapturedwithinthegameitsdf.For e..x.ample,thewinnerof thechampionshipeventof theWorldSeriesof Pokerisvirtually guaranteedtoreapawindfallfromendorsements, appearances,andsoon,over and above the:large firstprize. Thereareotherconsiderationsforplayersatthepokertableaswell;perhapslosingan additionalhand wouldbeasignificant psychologicalblow."Whilewemay criticizethis view asirrational,itmuststillfactorineoanyexhaustiveexaminationof theincentivestoplay poker.Even if we restrict our inquiry eomonetary rewards, we findthat preference for money isnon-linear.For most people, winnlng fivemillion dollars isworth much more(or has much moreutility)than a50% chance of winning tenmillion;fivemilliondollarsislife-changing money for most,and themarginal value of the additional fivemillion ismuch smaller. In abroader sense,allof theseissuesareincluded in theutility theory branch of economics. Utility theorists seek to quantify the preferences of individuals and create a framework under whichfinancialandnon-financialincentivescanbe directly compared.In reality, it isutility that we seek to maximize when playing poker (or in fact,when doing anything). However,the use of utility theory asabasisforanalysispresentsadifficulty; each individualhashisown utility curves and so general analysis becomes cxtremdy difficulL In thisbook,wewillthereforerefrainfromconsidering utilityand insteadusemoney won insidethe game asaproxy forutility.In thebankroll theory section in Part IV, wewill take an in-depth look at certain meta-game considerations, introduce such concepts asrisk of ruin, theKelryc r i t e r i ~andcertaintyequivalent.Allofthesearemeasuresof riskthathave primarily to do with factors outside the game. Except when expressly stated, however, we will take as a premise that players are adequately bankrolled for the games they are playing in, and that their sole purpose isto maximizethe money they will -winby making the best decisions at every point. Maximizing total money won inpoker requiresthat aplayer maximize the expected value of hisdecisions.However,beforewecanreasonablyintroducethiscornerstoneconcept,we mustfirstspendsometimediscussingtheconceptsofprobabilitythatunderlieit.The followingmaterialowesagreatdebttoRlchardEpstein'stextTheTheory0/ Gamblingand StaJistiud Lo[jc (1967),avaluable primer on probability and gambling. THE MATHEMATICS OF POKER13 Part I:Basics Probability Most of thedecisionsin poker take place under conditionswheretheoutcomehasnot yet beendetermined.Whenthedealerdealsoutthehandattheoutset,theplayers'cardsare unknown,atleastuntiltheyareobserved.Yetwestillhavesomeinformationaboutthe contents of the other players' hands. The game's rules constrain the coments of their hands-while a player may hold the jack-ten of hearts, he cannot hold the ace-prince of billiard tables, forexample.Thecompositionofthedeckof cardsissetbeforestartingandgivesus information about the hands. Consider a holdem hand. "What isthe chance that the hand contains two aces? You may know theansweralready,but consider what theanswer means."What if wedealtamillion hands just likethis? How many pairsof aceswouldthere be? What if wedealttenmillion? Over time and many trials,the ratio of pairs of aces to total hands dealt v.'ill converge on a particular number.Wedefine probabilii)Jasthisnumber.Probabilityisthe keytodecision-making in poker asit providesamathematicalframeworkby whichwecanevaluatethelikelihoodof uncertain events. IT n trials of an experiment (such asdealing out a holdem hand)produce no occurrences of an event x,wedefinethe probability p of x occurringP(x)asfollows: limn, p(x} n-".n (1.1 ) Now it happens to be the case that the likelihood of a single holdem hand being a pair of aces is 11 221 We COUld, of course, determine this by dealin'b out ten billion hands and obscrving,the ratlo01palls01acesto tota\. banusuea\t. l:"rU.s,'however, wou\e be a \eng\hyane Oillicu\t process, and we can do better by breaking the problem up into components. First we consider just one card.What isthe probability that a singlecard isan ace?Even thisproblem can be broken down further - what isthe probability that a single card isthe aceof spades? This final question can be answered rather directly.Wemakethefollowing assumptions: There are fifty-twocards in a standard deck. Each possible card isequally likely. Then the probability of any particular card being the one chosen is1/52,If the chance of the cardbeingtheaceof spadesis1/52,what isthechanceof thecardbeing anyace?Thisis equivalent to the chance that the card isthe ace of spades OR that it isthe ace of hearts OR that it istheaceof diamondsOR that it istheaceof clubs.There arefouracesin thedeck, each with a 1/52chance of being the card, and summing these probabilities,wehave: 1 P(A) =13 Wecan sum these probabilitiesdirectly becausethey aremutually exclusive;that is, nocard can simultaneously be both the ace of spadesand the ace of hearts.Note that the probability 1/13 isexactlyequaltotheratio(number of acesinthedeck)/(number of cardstotal). This rdationship holds just aswellasthe summing of the individual probabilities. 14THE MATHEMATICS OF POKER et re 1e :5, lS w Is :r U-n )f n s e Chapter 1-Decisions Under Risk:Probability andExpectation Independent Events Some events,however, are not murually exclusive.Consider forexample, these two events: 1.The card isa heart 1.The card isan ace. If we try to figure out the probability that a single card isa heart OR that it isan ace,wefind :hat there are thirteen hearts in the deck out of fifty-cards, so the chance that the card is a heart is The chancethat thecard isan ace is: asbefore,1/13_ However, wecannot simplyadd :heseprobabilitiesasbefore,because it ispossible fora card to both be an ace and aheart. There aretwotypesof relationshipsbetween events.The firsttypeareeventsthathaveno effectoneach other.Forexample,theclosingvalueof theNASDAQ stockindexandthe value of the dice on a particular roll at the crapstable in a casino in Monaco that evening are basicallyunrelatedevents;neitheroneshouldimpacttheotherinanywaythatisnot If theprobabilityof botheventsoccurring equalstheproduct of theindividual probabilities,thentheeventsare saidto be 'independent. The probability that both A and B cur iscalled the joint probability of Aand B. :::. :hiscase,the joint probability of acardbeing both aheart and anace is(1 /13)(1/4),or 1/52. :-- ,isbecause the factthat the card is a heart does not affect the chance that it isan ace - all i:r= suitshave the same set of cards. :::iependent events are not murually exclusive except when one of the events has probability .::::::0.Inthisexample,the total number of heartsin the deck isthirteen,and the total of aces :::J.the deck is four. However, by adding these together, we are double-counting one single card the ace of hearts) . There are acrually thirteen hearts and three other aces, or if you prefer, four aces, and twelve other hearts.Itrums out thatthe generalapplicationof thisconcept isthat the probability that at least one of two mutually nonexclusive events A and B will occur is the sum of the probabilities of Aand B minus the joint probability of Aand B.So the probability of the card being a heart or an ace is equal to the chance of it being a heart (1/4) plus the chance of itbeinganace(1/13)minusthe chanceof itbeingboth(1/52),or 4h3. Thisistrueforall cyenu, iudcpcnde.nt or dependent. Dependent Events Some evenrs, by cono-ast, do have impacts on each other. For example, before a baseball game, acertaintalentedpitchermighthavea3% chanceof pitching nineinningsandallowingno runs, while histeam might havea60%chanceof wllming thegame. However,the chance of thepitcher'Steamvvinningthegameandhimalsopitchingashutout isobviouslynot60% times3%. Instead, itisvery closeto3% itself, because the pitcher'steam will virtually always "in thegamewhenheaccomplishesthis.TheseeventsarecalleddependentWecanalso consider the ronditUmal probabiliry of A given B, which isthe chance that if B happens, A will alsohappen.The probability of Aand B both occurring fordependent eventsisequalto the probability of A multiplied by the conditional probability ofB given A.Events are independent if the conditional probability of Agiven B isequal tothe probability of A alone. Sununarizing these topics more formally,if we use the following notation: p(AU B) =Probability of A or B occurring. p(An B) =Probability of A and B occurring. THEMATHEMATICS OF POKER15 I Part I:Basics peA I B) ~Conditional probability of A occurring given B hasalready occurred. The U and n notations are from set theory and fonnally represent "union" and "intersection." Weprefer the more mundane terms"or" and "and." Likewise, I isthe symbol for"given,"so we pronounce these expressions as follows: p(AU B)~"pof Aor B" p(An B) ~"p of AandB" PtAI B)~"p of AgivenB" Then formutually exclusive events: p(AU B)~prAY+ p(B) For independent events: p(An B)~p(A)p(B) For all events: p(AU B)~prAY+ p(B)- p(An B) For dependent events: p(An B)~p(A)p(B I A) (1.2) (1.3) (1.4) (1.5) Equation 1.2issimply a special case of Equation 1.4 for mutually exclusive events, p(An B)=O.Likewise,Equation1.3isaspecialcaseof Equation1.5,asforindependent events, p(BI A) ~p(B).Additionally, if p(BIA) ~p(B),then peAIB)~peA) We can now return tothe question at hand. How frequently will a single holdem hand dealt froma full deck contain two aces? There aretwo events here: A:The first card isan ace. B:The second card isan ace. p ( A ) ~'113, and p ( B ) ~' 113as well. However, these two events are dependent.lf A occurs (the first cardisan ace),then itislesslikelythat B will ocrur, asthe cards are dealt without replacemem. So P{B I A)isthe chance that the second card isan ace given that the first card isan ace. There are three aces remaining,and fifty-onepossible cards, so p(B IA)=3/51,or 1/17. p(An B)~p(A)p(BI A) p(An B)~('I.,) (II,,) p(An B)~'I", Tnere are a number of ol.her simple propertiesthat wecan mention about probabilities.First, theprobability of anyevent isatleastzeroandno greaterthanone.Referring backtothe definition of probability, ntrials will never result in more than n occurrences of the event, and never lessthanzerooccurrences. The probability of an eventthatiscertain tooccur isone. The probability of an event that never occurs is zero. The probability of an evenes complement -thatis,thechancethataneventdoesnOtoccur,issimplyoneminustheevent's probability. Summarizing, if weuse the foUowingnotation: P(A) ~Probability that A doesnot occur. 16THEMATHEMATICS OF POKER Chapter l-Decisions Under Risk: Probability andExpectation C = a certain event 1= an impossible event -=nenwe have: o ""P(A)"" 1forany A (1 .6) p(C)~1(1.7) prJ)~0 (1.8) p(A)+ PtA) ~1(1 .9) Equation1.9can alsobe restated as: p(A) ~1 - PtA)(1.10) \'e can solve many probability problems using these rules. Some common questions of probability are simple, such asthe chance of rolling double sixes on [wodice.In tenns of probability,thiscan bestatedusing equation 1.3, since the dierolls areindependent.Let PtA)bethe probability of rollingasixonthefirstdieand P(lJ)be the probability of rolling a six on the second die. Then: p(A n B)~p(A)p(B) p(An B)~(110)('10) p(An B)~1/" Likewise,usingequation1.2,mechanceof asingleplayerholdingaces,kings,orqueens becomes: P(AA)~I/m P(KK)~1/221 P(QQJ ~1/221 PAA,KK,QQ,J)~P(AA)+ P(KK)+ P(QQJ ~J/22l Additionally we can solve more complex questions, such as: How likely is it that a suited hand will flopa Bush? \Vehold twoof the flush suit, leaving eleven in the deck. All three of the cards must be of the Bushsuit,meaning thatwehaveA =thefirstcardbeing aflushcard,B =thesecondcard being a Bush card given that the firstcard isa Bushcard,andC ~the third card being a Bush card given than both of the firsttwoare flushcards. prAY~II/SO p(BI A) ~10/" p(e I (An B)) ~'/48 (two cardsremoved fromthe deck in the player'shand) (one flushcard and threetotal cards removed) (two flush cards and fourtotal cards removed) Applying equation 1.5,we get: p(An B)~p(A)p(BI A) p(An B)~(11/50)(,0/,,) p(An B)~Il/"s THE MATHEMATICS OF POKER17 Part I: Basics Letting D= (An B),we can use equation1.5 again: p(D n C)p(D)(P(CI D) p(An B n C)p(An B)p(CI (An B)) p(An B n C)(11/24.;)('/,,) p(An Bn C)=33h920,or alittlelessthan 1%. vVecan apply these rules to virtually any situation,and throughout the text we will use these propertiesand rulesto calculate probabilitiesfor single events. Probability Distributions Though single event probabilitiesare important, it isoften the case thatthey are inadequate tofullyanalyzeasituation.Instead,itisfrequendyimportanttoconsidermanydifferent probabilities at the same time. We can characterize the possible outcomes and their probabilities froman event asa probability distributWn. Considerafaircoinflip.Thecoinfuphasjusttwopossibleoutcomes- eachoutcomeis mutually exclusiveand hasa probabilityof l /2 .Wecancreateaprobabilitydistributionfor the coin flipby taking each outcome and pairing it with itsprobability. So we have two pairs: (heads, II,)and (tails, II,) . If C isthe probability distribution of the result of a coin flip, then wecan write this as: {(heads,II,),(tails,II,)} Likewise,the probability distribution of the result of a fairsix-sided die roll is: (2,1/6) , (3,IIe),(4,1/6),(5,1/6),(6,110)) Wecanconstructadiscreteprobabilitydisrributionforanyeventbyenumeratingan exhaustive and murnally exclusive list of possible outcomes and pairing these outcomes "vith their corresponding probabilities. We can therefore create different probability distributions from the same physical event. From our die roll wecould also create a second probability distribution, thisone the distribution of the odd-or-evennessof the roll: {(odd,1/,),(even,II,)} In poker,wearealmostalwaysveryconcernedwiththecontentsof our opponents'hands. But itisseldom possibletonarrow down our estimateof thesecontentstoasinglepairof cards. Instead, we use a probability distribution to represent the hands he could possibly hold and the corresponding probabilitiesthat he holdsthem. At the beginning of the hand, before anyone haslookedattheir cards,each player' s probability distributionof handsisidentical. Asthe handprogresses,however,wecan incorporate new information wegainthroughthe play of the hand, the cards in our own hand, the cards on the board, and so on, to continually refine the probability estimates we have for each possible hand. Sometimes we can associate a numerical value with each element of a probability distribution. Forexample supposethat afriendoffersto flipa faircoin with you. The winner will collect $10fromtheloser.Nowtheresultsof thecoinflipfollowtheprobabilitydistributionwe identified earlier: 18THE MATHEMATICS OF POKER = 01 " n b te nt os Chapter 1-Decisions Under Risk:Probability andExpectation c = [(heads,If,) ,(tails,If,)) 5:'nce we know the coin is fair, it doesn' t matter who callsthe coin or what they call, so we can Xienrify a second probability distribution that isthe result of the bet: C' =[(win,If, ),(lose,If,)) '\'(:can then go further,and associateanumerical value -...vitheachresult.If we win the flip, our friendpaysus$10.If welosethe flip, then we pay him $10.Sowe have the following: B =[(+$10,If,) ,(-$10,I/,)} \ , n enaprobabilitydistributionhasnumericalvaluesassociatedwitheachof thepossible outcomes, we can find the expected value (EV) of that distribution, which isthe value of each outcome multiplied by itsprobability, all sununedtogether. lbroughout the text, we will use :he notation to denote "the expected value of X " For thisexample, wehave: =(1/,)(+$10)+(1/,)(-510) = $5 + (-$5) =O Hopefullythisisintuitivelyobvious- if youfupafaircoinforsomeamount,half thetime you win and half the time youlose. The amounts are the same, so youbreak even on average. _-\lso,theEV of decliningyourfriend' sofferby notHippingat allisalsozero,because no money changes hands. For a probability distribution P,where each of the n outcomes has a value Xiand a probability Pi. then P'J expected value is : (1.11 ) At the core of winning at poker or at any type of gambling isthe idea of maximizing expected value.In thisexample, your mend hasoffered you afair bet.On average,youare no better or worseoffby flipping with him than youare by declining to flip. )low suppose yourfriendoffersyouadifferent,betterdeal.He'llfupwithyouagain,but when you win,he'll pay you $11, while if he wins, you' llonly pay him $10. Again, the EV of not flipping is0,but the EV of flippingisnotzeroanymore.You'llwin $11when youwin but lose $10 when you lose.Your expected value of thisnew bet Bnis: =(' /,)(+$11)+(1/, )(-$10) =$0.50 On averagehere,then,youwill winfiftycentsper flip.Of course,thisisnot aguarameed win;infact,it' simpossibleforyoutowin 50centsonanyparticularflip.It' sonlyinthe aggregatethat thisexpectedvaluenumber exists.However,bydoing this, youwillaverage fiftycents better than declining. As another example, let's say your same mend offers you the following deal.You'llroll a pair of dice once,and if the dicecome up doublesixes,he'll pay you $30,whileif theycome up any other number,you'll pay him $1.Again,we can calculate the EV ofthis proposition. THE MATHEMATICS OF POKER19 Part I:Basics =(+$30)(1/36)+ =$30/" =_$5/36 or about 14 cents. The value of thisbet toyou isabout negative 14 cents. The EV of not playing iszero, so this isabad bet and you shouldn't take it.Tellyour friendtogoback tooffering you11-10on coin fups.Notice that this exact bet isoffered on craps layoutsaround the world. Averyimportantpropertyof expectedvalueisthatitisadditive.Thatis,theEV of six different bets in a row isthe sum of the individual EVs of each bet individually. Most gambling games - most things in life,in fact,are just likethis.Wearecontinually offered little coin fups ordicerolls- somewithpositiveexpectedvalue,otherswithnegativeexpectedvalue. Sometimestheeventinquestionisn't adierollor acoinffip,butan insurancepolicyora bond fund.The freedrinksand neon lightsof LasVegasarefinancedbythesummation of millions of little coin flips,on each of which the house hasa tiny edge. A skillful poker player takesadvantage of this additive property of expected value by constantly taking advantage of favorableEV situations. In using probability distributions to discuss poker, we often omit specific probabilities for each hand.When we do this, itmeansthat the relative probabilities of those hands are unchanged from their probabilities at the beginning of the hand. Supposing that we have observed a very tight player raise and we know from our experience that he raises if and only if he holds aces, kings, queens,or ace-king,we might represent hisdistribution of hands as: H =(AA,KK,QQ, AKs,AKo) The omission of probabilities here simply implies that the relative probabilities of these hands are asthey were when the cards were dealt. We can also use the notation forsituations where wehave morethanone distribution under examination. Suppose weare discussing a pokersituationwheretwoplayersAandBhavehandstakenfromthefollowing distributions: A =(AA,KK, QQ, lJ, AKu,AKs) B= (AA,KK,QQ} Wehave the following,then: :the expectation for playing the distribution A against the distribution B. :theexpectationforplayingthedistributionAagainstthehandAA taken fromthe distributionB. :the expectation forplaying AA from A against AAfromB. =P(AA) + p(KK) + P(Q9J ... and so on. Additionally,wecanperfonnsomebasicarithmeticoperationsontheelementsofa distribution.For example,if wemultiply allthe valuesof the outcomesof a distribution by a real constant,theexpected value of the resulting distribution isequal to the expected value of the original distribution multiplied by the constant. Likewise, if we add aconstant to each of the values of theoutcomes of a distribution, the expected value of the resulting distribution is equal to the expected value of the original distribution plus the constant. 20THE MATHEMATICS OF POKER :lis Dn ax :>g ps ~ e . a Df h d Y ;, s s , , Chapter l-DecisionsUnder Risk:Probability andExpectation ' . \ ~shouldalsotakeamomenttodescribeacommonmethodof expressingprobabilities, edds.Oddsaredefinedastheratioof meprobabilityof theeventnothappeningtothe :zobability of the event happening. These odds may be scaled COany convenient base and are ..:ommonly expressed as "7 to 5,""3to2,"etc.Shorter odds are those where the event is more ...1.dy: longer odds arethose wherethe event islesslikely.Often, relative hand valuesmight .x e.....'Pressed this way:"That hand isa 7 to 3 favorite over the other one," meaning that it has !j"00!0of winning, and so on. Oddsare usually more awkward to use than probabilities in mathematical calculations because ~ - cannot be easily multiplied by outcomestoyield expectation.True "gamblers" often use .::.dds. because odds correspondto the ways in which they are paid out on their bets. Probability ~:nare of a mathematical concept. Gamblers who utilize mathematics may use either, but often ;:rderprobabilitiesbecause of the ease of converting probabilities [0 expected value. Key Concepts Theprobability of an outcome of anevent is theratioof that outcome's occurrence over an arbitrarily large number of trials of that event. Aprobability distribution is a pairing of a list of complete andmutually exclusive outcomes of an event with their correspondingprobabilities. Theexpected valueof a valuedprobability distributionis the sumof the probabilities of the outcomes times their probabilities. Expected value isadditive. If each outcome of a probability distribution is mapped to numerical values,the expected value of the distribution isthe summation of t he products of probabilities and outcomes. Amathematical approach to poker isconcernedprimarily with the maximizationof expectedvalue. THEMATHEMATICS OF POKER21 Part I:Basics Chapter 2 Predicting theFuture: VarianceandSampleOutcomes Probability distributions that have values associated with the clements have two characteristics which,takentogether,describemOStof thebehaviorof thedistributionforrepeatedtrials. The first,describedinthepreviouschapter,isexpectedvalue.Thesecondisvariance,a measure of thedispersionof theoutcomesfromtheexpectation.Tocharacterizethesetwo tcrtIl51oosely, expected value measures how much you will win on average; variance measures how far your speciEc resultsmaybe fromtheexpected value. When we consider variance,weare attempting to capture the range of outcomesthat can be expectedfromanumberof trials.Inmanyfields,therangeof Outcomesisof particular concern on both sides of the mean. For example, in manymanufacturing environments there isa band of acceptability and outcomes oneither side of thisband are undesirable.In poker, there isa tendency to characterize variance as a one-sided phenomenon, because most players are unconcerned with outcomes that result in winning much more than expectation.In fact, "variance" isoften used asshorthandfor negative swings. This viewissomewhat practical,especially for professional players, but creates a tendency to ignorepositiveresultsandtothereforeassumethatthesepositiveoutcomesaremore representative of the underlying distributionthan they really are.One of the important goals of statisticsistofindtheprobabilityof acertainmeasuredoutcomegivenasetof initial conditions,andalsotheinverse of this- inferringtheinitialconditionsfromthemeasured outcome. In poker, both of these are of considerable use. We refer to the underlying distribution of uutCOInesfrUInasetof initial conditionsasthe population and the observedoutcomes as thesample.In poker,we oftencannot measurealltheclements ofthe population, but must content ourselves with observing samples. Most statisticscoursesandtextsprovidematerialonprobabilityaswellasawhole slewof samplingmethodologies,hypothesistestS,correlationcoefficients,andsoon.In analyzing pokerwemakeheavyuseof probabilityconceptsandoccasionaluseof otherstatistical methods.Whatfollowsisaquick-and-dirtyoverviewof somestatisticalconceptsthatare usefulin analyzing poker, especially in analyzing observed results.Much information deemed tobeirrelevantisomittedfromthefollowingandweencourageyoutoconsultstatistics textbooks formore information on thesetopics. Acommonlyaskedquestioninpokeris"HowoftenshouldIexpecttohaveawinning session?"Rephrased,thisquestion is"what is the chancethat a particular sample taken from apopulation that eonsistsof my sessionsin agame will havean outcome> O?"TIl.emost straightforwardmethodofansweringthisquestionwouldbetoexaminetheprobability distribution of your sessions in that game and sum the probabilitiesof all those outcomes that are greater than zero. Unfortunately,wedo not haveaccesstothat distribution - no matterhowmuch datayou have collected about your play in that game fromthe past, all you have isa sample. However, suppose that we know somehow your per-hand expectation and variance in the game, and we know how long the session you are concemed with is. Then we can use statistical methods to estimate rhe probability that you will have a winning session. The first of these items, expected value(which we can also callthe mean of the distribution) isfamiliar bynow;we discussed it in Chapter 1. 22 THE MATHEMATICS OF POKER 5 Chapter 2-Prediding the Future: Variance andSample Outcomes Variance The second of these measures, variance, isa measure of the deviation of outcomesfromthe expectationof adistribution.Considertwobets,one whereyou are paid evenmoneyon a coin flip,and one where you arepaid 5to1 ona die roll, winning onlywhenthe die comes up6. Both of thesedistributionshaveanEVof 0,butthedierollhassignificantlyhigher \w ance.lh of the time,yougeta payout that is5unitsawayfromthe expectation, while 5/6 ofthetimeyougetapayoutthatisonly1unitawayfromtheexpectation.Tocalculate variance,wefirstsquarethedistancesfromtheexpectedvalue,multiplythembythe probability they occur, and sum the values. For a probability distribution P, where each of the n outcomes has a value Xi and a probability p, . then the variance of P,Vpis: Vp ~I,p,(x,- < P ' (2.1) ; =1 );"otice that because each teIm issquared and therefore positive, variance is always positive. Reconsidering our examples,the variance of the coinHipis: Vc=(1/,)(1- 0)' +(1/,)(- 1 - 0)' Vc= 1 ' Vhilethe variance of the die rollis: VD =('10)( 1 - 0)' +(1/6)(5 - 0)' VD= 5 In poker, a loosewild game will have much higher variance than a tight-passive game, because me outcomes will be further fromthemean(pots you win will be larger, but themoney lost in potsyoulose will be greater). Style of play willalsoaffectyour variance; thin value betsand semibluff raises are examples of higher-variance plays that might increase variance, expectation, or both.On me orner hand, loose-maniacal players may make plays that increase their variance whiledecreasing expectation.Andplaying too tightly mayreduce both quantities. In Part IV, wewill examine bankroll considerations and risk considerations and consider a framework by which variance can affect our utility value of money. Except forthat part of thebook,wewill ignore variance asa decision-making aiterion forpoker decisions.In thisway variance isfor usonly a descriptivestatistic, not a prescriptive one (asexpectedvalue is). Variancegivesusinformation abouttheexpecteddistancefromthemean of adistribution. Themostimportant propertyof varianceisthatitisdirectlyadditiveacrosstrials, justas e:\'}Jectationis.Soifyoutaketheprecedingdicebettwice,thevarianceof thetwobets combined istwiceaslarge,or10. Expected value is measured in units of expectation per event; by contrast, variance ismeasured in units of expectation squared per event squared. Because of this, it isnot easyto compare variancetoexpectedvalue directly.If we are to comparethese two quantities,we musttake the square root of the variance, which iscalledthe .standard deviation.For our dice example, thestandard deviationof onerollis{5:::::2.23.WeoftenusetheGreek letter0(sigma)to representstandarddeviation,andbyextension02 isoftenusedtorepresentvariancein addition to the previously utilizedV. THE MATHEMATICS OF POKER23 -Part I:Basics (2.2) 0-2 =V(2.3) The Normal Distribution When we take a single random result from adistribution, it has some value that isone of the possible outcomes of the underlying distribution. We call thisarandom variable.Suppose we flipacoin.The flipitself isarandom variable.Supposethatwelabelthetwooutcomes1 (heads)and 0(tails). The result of the flipwill theneitherbe1(half thetime)or 0(half the time). If we take multiple coin flipsand sum them up,we get a value thac isthe summation of theoutcomesof therandomvariable(forexample,heads),whichwecallasample. The sample value,then,vvill be the number of heads we flipin whatever the sizeof the sample. For example, suppose we recordedthe resultsof 100coinHipsasa singlenumber - thetotal number of heads. The expected value of the sample "",illbe 50, aseach fliphasan expected value of 0.5. The variance and standard deviation of a single fupare: 02 ~(1 /2) (1 - ' /2)2+ (1/2)(0 - ' /2)' a2=1h a=1/ 2 From the previous section, we know alsothat the variance of the flipsisadclitive. So the variance of 100 flipsis25. Just asan individual fliphasa standard deviation, asample hasa standard deviation aswelL However,unlikevariance,standarddeviationsarenotadditive.Butthereisarelationship between the twu. For Ntrials,the variance will be: a2N =Na2 (2.4) The square root relationship of trialstostandard deviation isan important result, because it showsushowstandard deviationscalesovermultipletrials.If weflipacoinonce,ithasa standard deviation of 1/2.If we flipit100 times, the standard deviation of a sample containing 100 trials isnot 50, but 5, the square root of 100 timesthe standard deviation of one Hip.We can see, of course, that since the variance of 100 flips is25, the standard deviation of 100 flips issimply the square root,S. The distributionof outcomes of asampleisitself aprobabilityclistribution,and iscalledthe sampling distribuiWn. An important result from statistics, the Central Limit 7heorem, describes the relationship between thesampling distribution and the underlying distribution.'What the Central Limit Theorem saysisthat asthe sizeof thesampleincreases,thedistribution of the aggregatedvaluesofthesamplesconvergesonaspecialdistributioncalledthenormal distribution. 24THE MATHEMATICS OF POKER he Ne 1 he of he tal ,d ll. Ip j, a Ig {e JS Ie os Ie Ie 21 :R Chapter 2-Prediding the Future: Variance andSampleOutcomes The normal distribution is a bell-shaped curve where the peak of the curve isat the population mean,andthetailsasymptoticallyapproachzeroasthex-valuesgotonegativeorpositive infinity.The curveisalso scaledby the standard deviationof thepopulation.The total area under the curve of the normal distribution(aswith allprobability distributions) isequalto1, and the area under the curve on the interval [xl>x2]isequal to the probabili[}' that a particular result will fallbetween XIand x2'1bis area ismarked region A inFigure 2.1 . .4 0.35 .3 0.25 0.2 .1 5 0.1 0.05 X2 0 3-2-1023 Figure 2.1. Std. Normal Dist, A = p(event between XIand~ ) ...\littlelessformally,the Central Limit Theorem saysthat if youhave some populationand take a lot of big enough samples(how big depends on the type of data you're looking at),the outcomesof the sampleswill followa bell-shapedcurve aroundthe mean of the population \\ith a variance that' s related to the variance of the underlying population. The equation of thenormal distribution functionof a distribution with mean Jiand standard deviation 0"is: 1(x-)1. )' N (x,)1.,rr) =~ e x p (---,-) rrv2n2rr (2.5) Findingtheareabetweentwopointsunderthenormaldistributioncurvegivesusthe probabilitythatthevalueof asamplewiththecorrespondingmeanandvariancewillfall between those two points. The normal distribution is symmetric about the mean, so '/2 of the total area isto the right of the mean, and 1/2is to the left.A usual method of calculating areas under the normal curve involves creating for each point something called a 7--score,where z = 6 - Ji)/a.lbis zscorerepresentsthenumber of standarddeviationsthattheoutcomex is away fromthe mean. , ~ ( x- p}la(2.6) THE MATHEMATI CS OF POKER25 Part I:Basics We can thenfindsomething calledmecumulative normal distribution foraz-scoreZ,which isthe area to the left of z under the curve (where me mean iszero and the standard deviation is1). We callthisfunction (z).See Figure 2.2 If z isanormalized z-score value,then the cumulative normal distribution functionforz is: 1f'(,) =r;:- ,xp( - -) .,2"- 2 (2.7) 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 a 32 10 13 x Figure 2.2. Cumulative normal distribution Fmding the area between two values xl and x2 is done by calculating the z-scores ziand 1.2for Xland X2,finding the cumulative normal disttibution valuestD(Z\) and {z2) and subo-acting them. If (z)isthe cumulative nonnal distributionfunctionforaz-score of Z,thenthe probability that asampletaken from anormal distribution function ,"lithmean p and standard deviation 0"willfallbetweentwo z-scoresXland X2is: (2.8) Statisticianshavecreatedtablesof =Ih unitsltrial ""ben the player wins, he gains six units. Subtracting the mean value of 1/6 from this outcome, obtain: Vw;n =(6 - 'I.)' =(35/6)2 :.ike\\ise,when he loses he losesone unit. Subtracting the mean value of Ihfrom this, we have: Via," =( 1 - 'I,)' Vlose=("1.)' The variance of the game is: VD2 =p (win)(Vwin)+ p(lose)( fl o,e) VDl =(' /,)(35/,)'+ ('1.)(" 1.)' VD2z6.806units2ltria12 Suppose wetossthe die 200 times.What isthe chance that the player will "vin overall in 200 tosses? Will 40 unitsor more? Will 100 units or more? \ \e c;msolvethisproblem using thetechniqueswehave just summarized.Wefirstcalculate the standarddeviation,sometimescalledthe standard error,of asampleof 200trials.TIlis willbe: a= ff= --16.806 = 2.61unitsltrial .\pplying equation 2.4 we get : aN =a-.JN a200=2.61--1200=36.89 units /200trial,. For 200trials, theexpectedvalue,or mean(P),of thisdistribution is1/6units/trial times200 trialsor 33.33 units. Using Equation 2.6, we findthe z-score of the point 0 as: Ix={x- p)/crwherex =0 Zo=(0.33.33) 136.89 Zo= 33.33 / 36.89 Zo= 0.9035 THE MATHEMATICS OF POKER27 Part I: Basics Consultingaprobabilitytableinastatisticstextbook,wefindthattheprobabilitythatan observedoutcomewill lietothe leftof chisz-score, (-0.9035) is0.1831, Hence,there isan 18.31% chance that the player willbe behind after 200trials. Tofindthe chance of being 40 unitsahead, wefindthat point'sz-score: '40 (40 - 33.33)/ (36.89)0.1807 (0.1807)0.5717 But Cl> (O.l807)is the probability that the observed outcome lies tothe left of 40 units,or that welose at least 40units. Tofindthe probability that wearetotherightof thisvalue,or are ahead 40units, we must acrually find1 - (0.1807). 1 - (0.1807)1 - 0.5717 Sothere isa 42.830/0 chance of being ahead at least 40 units after 200tosses. And similarly for100 unitsahead: '100 (100 - 33.33)/(36.89)1.8070 From a probability table wefindthat (1.8070) 0.9646. Thus, for: p 1 - (1.8070) 0.0354 TIle probability of being 100 units ahead after 200tossesis3.54%. These values, however,areonly approximations;thedistributionof 200-rollsamplesisnot quitenormal.vVecanactuallycalrulatethesevaluesdirectlywiththeaidof acomputer. Doing thisyidds: i Direct Calculation! Nonnal Approx. Chance of being aheadafter 200trials:i 81.96% 81.69% Chance of being aheadatleast 40units:: 40.460/0i 42.83/0 Chance of being aheadatleast100 units :: 4.44%: 3.54% As you can see,these values have slight differences.Much of this is caused by the fact that the direct calculations have only a discrete amount of values. 200 trialsof thisgame can result in outcomesof +38 and +45, but not +39 or +42,because there isonlyone outcome possible from winning1 a +6. The normal approximation assumesthat all values are possible. Using chismethod,wererumtothequestionposedatthe beginning of thechapter: "How often'willIhaveawirmingsession?"Tosolvethisproblem,weneedtoknowtheplayer's CA-p(0.1299) 44.83% P 1 . (0.1299) 1- 0.4483 P 0.5517 1b.is indicatesthat the player has a result of 0 or less44.83% of the time - or correspondingly hasa wirming session 55.17% of the time.In reality,playersmay change their playing habits in order to produce more winning sessions for psychological reasons, and in reality "win rate" fluctuatessignificantlybasedongameconditionsandforotherreasons.Butaplayerwho simplyplays300handsat atimewiththeaboveperfonnancememesshouldexpecttobe above zero just over 55% of the time. A more exn-eme example: A player once told one of the authors that he had kept a spreadsheet where he marked down every AQvs. AK all-in preflop confrontation that he saw online over a period of monthsand that after 2,000 instances,they were running about 50-50. How likely is this,assuming the site's cards were fairand we have no additional information basedon the hands that were folded? Erst, let's consider the variance or standard deviation of a single confrontation. .AKvS. AQis about 73.5%all-inpreflop(including allsuitednesscombinations). Let's assign aresult of 1 toAK winning and 0to AQwinning. Then our mean is0.735.Calculating the variance of a single confrontation: (0.735)(10.735)' + (0.265)(00.735)' 0.1948 cr0.4413 The mean of 2,000 hands is: )i"N)i )i"(2000)(0. 735) )i"1470 THE MATHEMATICS OF POKER29 Part I:Basics For a sample of 2,000hands,using Equation 2.4,the standard deviation is: f J N ~ & fJ2000~(0.4413)('1 2000) fJ2000~19.737 The resultreportedherewasapproximately 50%of 2000,or1000, whilethesamplemean would be about 1470out of 2000. We can findthe z-score using Equation 2.6: ,~(x~?N)/fJ '1000 ~(1000-1470)/(19.737) '1000 ~-23.8 15 The resultreported in thiscasewas1000out of 2000, whiletheexpectedpopulationmean wouldbe 1470out of 2000.This result isthen about 23.815 standard deviationsaway from the mean.Valuesthis smallarenot very easyto evaluate becausethey are so small - in fact, my spreadsheet program calculates 1.15 p + (2)(1.61) > 1.15 J1> -2.07 THE MATHEMATICS OF POKER Part I:BasIcs SupposewehaveasampleNthatconsistsof16,900handstakenfromanunderlying distribution with amean, or ",,>inrate,of 1.15 BB/ IOOhand astandard deviation of2.1 BBIh. Then, using equation 2.4: aN a..fJi a16,900(2,1BB/h) hands) a16,900 273BB,so aN/ l00h 273/1691.61 The standard deviation of a sample of thissize isgreater than the win rate itself. Suppose that weknewthattheparameters of theunderlying distributionwerethesameastheobserved ones. If wetook another sample of 16,900 hands, 32% of the time,the observed outcome of the 16,900 hand sample wouldbe lower than -0,46 BB/ l00 or higher than 2,76 BBIlOO, This isalittle troubling.How can webe confident in the idea that thesamplerepresentsthe true population mean,when even if that werethecase,another samplewould be outside of even those fairlywide bounds 32% of thetime? And what if thetrue population mean were actually,say,zero? Then1.15wouldfallnicely intothe one-sigmainterval.In fact, itseems like we can'ttellthe difference very clearlybasedon thissample between awin rate of zero and awin rate of 1.15 BB/ IOO, ' -\Thatwe can doto help tocapture thisuncertainty is create aconfidence interval. To create a confidence interval,wemust firstdecideon alevelof tolerance.Because we're dealing with statisticalprocesses,wecan'tsimplysaythattheprobabilitythatthepopulationmeanhas somecertainvalueiszero- wemighthavegottenextremelyluckyorextremelyunlucky. However,wecanchoosewhat iscalledasignificancelevel.Tillsisaprobability valuethat represents our tolerance for error. TIlen the confidence intelVal isthe answer to the question, "What are all me population mean values such that the probability of the observed outcome occurring islessthan the chosen significance level?" Suppose mat for our obselVed player,we choose a significance level of95%. Then we can find a confidence level for our player. If our population mean is 1',then asample of thissizetaken fromthispopulation willbe between()I- 2u)and()I+ 2a)95% of therime,So wecan find all the values of I' such that the obselVed value x =1.15 isbecween these two boundaries. As we calculated above,the standard deviation of asample of 16,900 hands is1.61 units/100 hands: aN a..fJi a(2, 1 BE/h) (J273BBper16,900 hands a/ IOOh273BB/169 1.61 So aslong asthe population mean satisfiesthefollovvingtwo equations,it will be within the confidence interval: 34 ()I - 2a) < U5 ()I + 2a) > 1.15 ()I- 2a) < U5 )1 - (2)(1.61) < 1.15 )I < 4,37 ()I + 2a) > 1.15 I' + (2)(1.61) > 1.15 I' > -2,07 THE MATHEMATICS OF POKER ng Ih. lat ed of he of Te os ro ' a as y. at n, Ie .d n d 10 e Chapter 3-UsingAllthe Information:EstimatingParametersandBayes' Theorem So a95% confidence interval forthisplayer'swin rate(basedon tbe16,900 hand samplehe hascollected) is[-2.07 BBIJOO, 4.37BB/IOO]. This doesnot mean that histrue rate is 95% likely tolie on thisinterval.TIlls isa conunon misunderstandingofthedefinitionof confidenceintervals.Theconfidenceintervalisall valuesthat, if theywerethetruerate,thentheobservedratewouldbeinsidetherangeof values that would occur 95% of the time. Classical statistics doesn' t make probability estimates of parameter values - in fact,the classical view isthat the true win rate iseither in the interval or it isn't, because it is not the result of arandom event. No amount of sampling can make us sure or unsure asto what the parameter value is. Instead, wecan only make claims abom the likelihoodor unlikelihoodthalwe wouldhaveobservedparticulaJoutcomesif aparameter had aparticular value. The maximumlikelihoodestimateandaconfidenceintervalareusefultoolsforevaluating what information wecan gain from a sample. In this case, even though a rate of 1.15 BBI100 might look reasonable, concluding that this rate isclose to the true one isa bit premature. The confidenceintervalcan giveusan idea of how widetherangeof possiblepopulation rates might be. However, if pressed, the best single estimate of the overall win rate is the maximum likelihood estimate, which isthe sample mean of 1.15 BBI100 in thiscase. To this point,wehave assumedthat we had no information about the likelihood of different win rates - that is,that our eh-perimental evidence was the only source of data about what win rate a player might have. But in truth, some win rates are likelier than others, even before we take ameasurement. Suppose that you played,say,5,000 hands of casino poker and in those hands you won 500 big bets,arate of 10big bets per100 hands. In this case,the maximum likelihood estimate fromthe last section would be that your \vID rate was exactly that - 10 big bets pcr 100 hands. But we do have other information. We have a fairly large amount of evidence, both anecdotal and culledfromhandhistorydatabasesandthelikethat indicatesthatamong playerswho playa statisticallysignificantnumberof hands,thehighestwinratesarenear 3-4BB/ IOO. Eventhefewoutlierswhohavewinrateshigherthanthisdonotapproacharateof 10 BB/lOO.Since thisisinformationthat we haveheforewe even startmeasuring anything, we call ita priori information. In fact , if we just consider any particular player we measure to be randomly selected from the universeof allpoker players,there isaprobabilitydistribution associatedwithhiswin rate. Wedon'tknowthepreciseshapeof thisdistribution,of course,becausewelack.observed evidence of the entire universe of poker players. However, if we can make correct assumptions abouttheunderlyinga priori distributionof win rates,wecan makebetter estimatesof the parameters based on the observed evidence bycombining the two setsof evidence. Bayes' theorem In Chapter 2, westated the basic principle of probability(equation 1.5). P(An B ) ~P(A )P(BI A ) In thisform,thisequation allowsustocalculatethe joint probabilityof AandBfromthe probability of A and the conditional probability of B given A. However, in poker, we are often mostconcernedwithcalculatingtheconditionalprobabilityof B giventhatAhasalready occurred - for example, weknow the cardsin OUTQ\o'l'nhand (A ), and now we want to know THE MATHEMATICS OF POKER35 Part I:Basics how thisinfonnation affecrsthe cards in our opponentshand(B).What we arelooking for is the conditional probability of B given A. So we can reorganize equation1.5 to create a formula for conditional probability. This isthe equation we will referto asBayes' theorem: (BIA) = P(A n B) pP(A) (3.1) Recallthat wedefinedBasthe complement of B in Chapter 1; that is: P(B) = 1- P(B) P(B)t P(B) = 1 Wealready havethe definitions: P( A n B ) =P(A)P(B I A) Since weknow that B and Bsum to1, P(A)can be expressedasthe probability of A given B when B occurs,plusme probability of A given B when B occurs. Sowecan restate equation 3.1as: (BIA)=p(AIB)P(B) PP(A I B)P(B) + P(A I B)P(B) (3.2) In poker, Bayes'theorem allowsus to refineour j udgments about probabilities based on new informationthatweobserve.In fact,strongplayersuseBayes'theoremconstandyasnew infonnation appears to continually refine their probability assessments; the process of Bayesian inference isat the heare of reading hands and exploitive play,asweshall see in Part II. Aclassicexampleof Bayes'theoremcomesfromthemedicalprofession.Supposethatwe haveascreeningprocessforaparticularcondition.If aniIl{liviuualwilhtheconditionis screened,thescreeningprocessproperlyidentifiesthecondition80%ofthetime.Ifan individual without the conditionisscreened,the screening process improperly identifies him ashavingthecondition10%ofthetime.5%ofthepopulation(onaverage)hasthe condition. Suppose,then,thatapersonselectedrandomlyfromthepopulationisscreened,andthe screening returnsa positive result.What isthe probability that thisperson hasthe condition (absent further screening or diagnosis)? If you answered"about or alittle lessthan 800/0,"youwerewrong,but not alone.Studies of doctors'responsestoquestionssuchasthesehaveshownaperhapsfrighteninglackof understanding of Bayesian concepts. We can use Bayes'theorem tofindthe answer to thisproblem asfollows: A =the screening process reUlms a positive result. B =the patient has the condition. 36THE MATHEMATICS OF POKER ;for n B ew ew an ve is n E 0 CO 0.05 0.1 0. 15 -0.2 o0.020.Q40.060.080. 1 Bluffing frequency Figure 4.1. Game Equity for various B strategies Figure 4.1showsthe linear functionsthat represent B'sstrategy options. We can see that B's maximallyexploitivestrategy involveschoosing thestrategythat hasthehigher value at the x-,-alue that A is playing. So when xis below 0.04, (meaning that A is bluffing 4% of the time), B should simply fold; above that B shouldcall.One thing that isimportant tonote about this isthatexploitiveplayofteninvolvesshiftingstrategiesratherdrastically.If Achangeshis x-valuefrom0.039co0.041,thatis,bluffingtwothousandthsof apercentmore often,B changeshis strategy from folding 100% of the timeto calling 100%of the time. Overthenextseveralchapters:wewilllookatsomeof theprinciplesof exploitiveplay, including pot odds and implied odds, and then consider the play of some example hands. vVe willconsider situations where the cardsare exposed but the play isnon-trivial and then playa single hand against adistribution of hands. We focusprimarily on the process of Dying to find the maximally exploitive strategy in lieuof giving specific play advice on any given hand, and especiallyon the process of identifYing specific weaknesses in the opponent's strategy. This last isparticularlyvaluable asit isgenerally quite repeatable and easy to do at the table. Pot Odds )lone of the popular forms of poker are static games (where the value of hands does not change from street to street). Instead, one common element of all poker games played in casinos isthe idea of the draw.VVhenbegUUlersare taught poker,they often learnthat four cards to aflush andfourcardstoastraightaredrawsandthathandssuchaspairs,trips,andBushesand straightsaremade Ium.ds.This isauseful simplification,but we use"draw" to mean avariety of types of hands. Most often, we use "draw" to refer tohands whose value if the hand of poker endedimmediatelyisnotbest amongthehandsremaining,but if certaincards(oftencalled outs)come,they will improvetobe best.However,in some cases,thiscan be misleading.For THE MATHEMATICS OF POKER49 Part II: ExploitivePlay example, consider the follovvingtwo hands on a flopofT+ 9+ 2+in holdem: HandA:Q+J+ HandB: A+3. Hand Ahasmore thana70% chanceof winning the hand despite hisfive-cardpoker hand being worse at thispoint. In thishand, it may seem a litde strange to refer to the hand that is more than a7 to 3 favoriteas"thedraw" whilethe other hand isthe"made hand,"because of the connotations we usually associate \viththese terms.However,we will consistently use theterm "draw"throughoutthisbook tomeanthe hand thatatthemoment hasthe worse five-cardpoker hand and needsto catch one of its~ U e s ,no matter how numerousthey may be, in order to win.Incontrast,wewill usethetenn"favorite"tomeanthat hand thathas more equity in the pot and "underdog"to mean the hand that hasless. In the above example, the Q+ J+ hand isboth the favoriteand a draw. One of me fundamental confrontations bet\'veen types of hands in poker isbetween made hands anddraws.Thisconfrontation isparticularly accentuated in limit poker, where themade hands areunabletomakebetslargeenoughtoforcethedrawstofold;instead,theysimplyextract value,whilethe drawscallbecause the expectationfromtheir share of thepot ismore than the amoum they must call. However, all is not lost in big-bet poker for the draws. As we shall see later on, draws are able to take advantage of the structure of no-limit gamesto create a special type of siruation that isexrremely profitable by employing a play called the semi-blt!IJ Example 4.2 The game is$30-60 holdem.Player A hasA+A. Player B has9. 8. The board is K. 7+ 3+2. The pot is$400.Player A isfirst.How should theaction goif both playersknow the full situation? Youcan likely guessthat the action goesA bets- B calls.It isvaluable, however, to examine theunderlyingmathematicsbecauseitprovidesanexcellentintroductiontothistypeof analysis and to the concept of pot odds. If PlayerAchecks,thenPlayerBwill.certainlycheckbehind.If Bweretobet,hewould immediately lose at aminimum 3/5of hisbet(becauseheonly winsthe pot1/5of thetime), plus additional EV if Player A were to raise. There will be no betting on the river,since both players will know who haswon the hand. Rather, any bet made by the best hand will not be called on the river, so effectively the pot will be awardedto the best hand. Since 35 of the remaining 44 cardsgive AAme win, weuse Equation 1.11to determine its EV from checking to be: =(P(Awins))(pot size) =("/,,)(400) =$318.18. Now let's considerB's options. Again,B will not raise,ashe has just alh chance of winning the pot (with 9 cards out of 44) and A will never fold.So B must choose between calling and folding. 50 =(P(B wins)) (new pot size) - (COstof acall) =(9/ ,,)($400+60+60)- $60 =(9/,,)($5 20) - $60 =$46.36 =0 THE MATHEMATICS OF POKER E::.ererrnine A's oc.e of winning oo:ncalling and CS OF POKER Chapter 4-PlayingtheOdds: PotOddsandImplied( Since B's EVfrom calling ishigher than his EVfromfolding,he will call in response to ::urn A.That makes Jfs EV frombetting: =(P(A wins)) (new pOlsize)- (cost of abet) =(35/4,) ($520)- $60 =$353.64 Comparing Jfsequity frombetting toA'sequityfrom checking,weseethat Againsa =:lOrethan$35bybetting.Thisresultindicatesthemain principleof made hand vs . . .:onfrontations: In made hand vs.draw siJuaWms,tire made hand usually bets. Therearesomeexceptionstothisrule,asweshallseelaterwhen weconsidersome : .:nmplicateddraw vs.madehandgames.Now letusconsiderB'splay.Wefoundtha :quiry [rom calling was greater than the zero equity fromfolding. We can solve an ineql :0 findallthe equities at which B would call, letting xbe B'schance of winning the pot: B's chance of winning thepot)(new pot size) - (enst of a call) >0 520x- 60 > 0 x> 3126 So B ought to call if his probability of winning the hand is greater than 3126 ,which they, ~givencase. Aswe discussed in Part I,we might also say that the odds of B winning ::~p (win on turn)($135+$60).$30 ~8/4,($135+$60).$30 ~$4.67 ~p(win onriver)($195+$120).$60 ~(8/ 44)($315).$60 ~$2.73 (HereweignorethecasewhereAhitsanaceorkingonthetumandwinsthehand immediately. Since B can't call when we ignore this case, his equity isworse when we include those hands, and hence he can't call if weinclude it.) This posesadifficulty withour previouscharacterization of pot odds.Westated earlier,"B should call if hisodds of winning the pot are shorter than his pot odds?' But here B'schance of winning the pot by the end of the h;mrlis p(B wins) ~1 - p(B loses) By applying Equation 1.3: p(Bwins) ~1 - [p(B loseson tum)][p(B loses on river)1 p(B wins) ~1 - ('''/'5)(36/..) p(B wins)~0.327or 32.7% Here we use the joint probability PIAn B) ~PIA )P(B) ofB not winning the pot and subtract itfromonetofindtheprobabilityof him winningthepot.We couldlikewisecalculatethe chance of him winning on the tum and also on the river, and then subtract the chance that he wins on both streets. However, the single step above ismore straightforward forthistype of calculation. If B has nearly a one-third chance of winning the pot then why can we not call in all of these cases,sinceineachof themBisgettinglongeroddsfromthepotthantwotoone?For example, in 4.3, the smallest POt of the three, B isgetting ($75+$60) to $60, or 2.25to1.The answer tothisquestion isthatonly situationswhereB winson the next street count forhis immediate POtodds. 52THE MATHEMATICS OF POKER aw be 1/45 on :1S to nd de ' B .ce lCt Je Ie of ,e )f 1e "' :R Chapter 4-PlayingtheOdds: Pot Oddsand ImpliedOdds Example 4.6 (Why giving yourself odds by raisingdoesn't work) Consider the case we lookedat previouslywhere the pot is $135 on the Hop. ITB calls the Bop, there is$19.5in thepot on thetum. If thedrawmisses,thenwhen Abets$60, B lacksthe oddstocallthe bet on thetum. But what if B raisestheflop instead? Now A"Willreraise, B .,ill call,and B will have odds to draw on the tum as well.It turns out that this is a losing play :or B comparedtosimply calling the Bop andfoldingthetunl.(If B hitsone of hisouts, he willbet and shut A out of mepot, sothe full house redraw doesn' t come into play.) \\'"ecan break B's expectation imo two dements. One ishis expectation ifhc wins on the tum, and the other isthe expectation whenthe hand goesto theriver.Let's call B winning on the rum card Bf and B not winning on the lurn card BI. < B, raise flop> = = = < B,raise flop> + P(B wins onthe tum) (pot ifB raises and A reraises) - (cost of the reraise) (8/45)($135 +2($30+$30+$30))- $90 -$34 p(Bdoesn't win on the tum)[ p(B wins on the river)(pot after the turn bets)-(cost of the tumbet)J 0' /4,)[(8/44)($3 15+2($60))- $60J $15.70 - $34 +$15.70 - $18.30 P(BJ1 ($135+2 ($30))- $30 $4.67 So raising the Bop, even though it makes it such that B has oddsto callthe tum and draw to hisBushagain,hasover $20lessexpectationcomparedtosimplycalling.In real-lifepoker, both players don' t have all the information. In fact,there will often be betting on future streets after the draw hits his hand. The draw can usethisfact to profitably draw with hands that are :::lotgetting enough immediate pot odds. Implied Odds In the previous section, we discussed the concept of pot odds, which isa reasonable shorthand for doing equity calculations in made hand vs. draw situations. We assumed that both players knew the complete situation. This assumption, however,does not hold in real poker. After all, the purported drawing player could be bluffing outright, could have been drawing to a different draw(suchasastraightor twopair draw),or couldhave hadthebest handallalong. A5a :-e5ult,the player with the made handoften hasto calla bet or two after the draw comesin. ,\nen we looked at pot odds previously, the draw never got any money after hitting his hand, since the formerl y made hand simply foldedto his bet oncethe draw came in. However,such a strategy in a game with concealedcards wouldbe highly exploitable.Asa result,the draw can anticipate extracting some value when the draw comes in. The combination of immediate odds and expected value from later streetsiscalledimplied odds. Example 4.7 Consider the foll owing example: Imaginethesiruationis just asin thegame previously(player Aholds A. K. and player B holds8+ 7. The HopisA. K ~4 .),but Player A doesnot knowforsurethac Player B is THE MATHEMATICS OF POKER53 Part II:ExploitivePlay drawing. Instead of trying to arrive at some frequency with which Player A will call Player B's betsoncetheBushcomesin,wewill simplyassumethat Awill callonebet fromB on all remaining streets. Earlier we solvedforB'simmediate pot oddsgiventhat the handswereexposed.However, in this game, B's implied odds are much greater. Assume that the pot is$135 before the flop. Now on the Bop, Abets $30. We can categorize the futureaccionintothree cases: Case1: The turn card is a flush card. In thiscase, B wins$195 in the pot from the flush, plus $240 from th