powerball lottery odds

7/30/2019 PowerBall Lottery Odds

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Durango Bill'sApplied Mathematics

Powerball Odds

How to Calculate the Probabilities for thePowerball Lottery

(Updated for the 59/39 game)

If the only thing you are interested in is the probability (odds) of winning the Powerball Jackpot, the Multi-StateLottery gives a concise table at their web site. We will give the same information here, but also show you how theseodds are calculated.

Game Rules

The numbers picked for the prizes consist of 5 white balls picked at random from a drum that holds 59 ballsnumbered from 1 to 59. The Powerball number is a single ball that is picked from a second drum that has 39 numbeanging from 1 to 39. If the results of these random number selections match one of the winning combinations on yoottery ticket, then you win something. You can also buy a Power Play Multiplier option. The multiplier has equal

odds of being a 2, 3, 4, or 5 which multiplies all the lower prize amounts by the multiplier amount. The exception ishat all four possibilities will pay $1,000,000 if you match the 5 white balls.

The changes in the rules that went into effect on Jan. 4, 2009 make it more difficult to win the Jackpot. This in turwill lead to somewhat higher average Jackpots which appears to be Powerballs answer to the very large Jackpotswhich sometimes develop in the Mega Millions lottery. For any given number of tickets in play, the probability thathere will be multiple winners is significantly decreased.

In any combinatorics problem where all possible outcomes are equally likely, the probability of a successful outcos determined by finding the number of successful combinations, and then dividing by the total number of allombinations. There are nine possible configurations that will win something in the Powerball Lottery. For each ofhese, the probability of winning equals the number of winning combinations for that particular configuration divide

by the total number of ways the Powerball numbers can be picked.

Powerball Total CombinationsSince the total number of combinations for Powerball numbers is used in all the calculations, we will calculate it f

The number of ways 5 numbers can be randomly selected from a field of 59 is: COMBIN(59,5) = 5,006,386. (See thmath notation page or Help in Microsoft's Excel for more information on COMBIN).

For each of these 5,006,386 combinations there are COMBIN(39,1) = 39 different ways to pick the Powerballnumber. The total number of ways to pick the 6 numbers is the product of these. Thus, the total number of equallyikely Powerball combinations is 5,006,386 x 39 = 195,249,054. We will use this number for each of the followingalculations.

ackpot probability/odds (Payout varies)The number of ways the 5 numbers on your lottery ticket can match the 5 white balls is COMBIN(5,5) = 1. Thenumber of ways your Powerball number can match the single Powerball number is: COMBIN(1,1) = 1. The producthese is the number of ways you can win the Jackpot: COMBIN(5,5) x COMBIN(1,1) = 1. The probability of succe

s thus: 1/195,249,054 = 0.000000005121663739+. If you express this as One chance in ???, you just divide 1 b

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he 0.000000005121663739+, which yields One chance in 195,249,054.

Match all 5 white balls but not the Powerball (Payout = $200,000)The number of ways the 5 numbers on your lottery ticket can match the 5 white balls is COMBIN(5,5) = 1. Thenumber of ways your Powerball number can match any of the 38 losing Powerball numbers is: COMBIN(38,1) = 38Pick any of the 38 losers.) Thus there are COMBIN(5,5) x COMBIN(38,1) = 38 possible combinations. The

probability for winning $200,000 is thus 38/195,249,054 = 0.000000194623222+ or One chance in 5,138,133.00.

Match 4 out of 5 white balls and match the Powerball (Payout = $10,000)

The number of ways 4 of the 5 winning numbers on your lottery ticket can match the 5 white balls is COMBIN(5,4). The number of ways the losing white number on your ticket can match any of the 54 losing white numbers isCOMBIN(54,1) = 54. The number of ways your Powerball number can match the single Powerball number is:COMBIN(1,1) = 1. The product of these is the number of ways you can win this configuration: COMBIN(5,4) xCOMBIN(54,1) x COMBIN(1,1) = 270. The probability of success is thus: 270/195,249,054 = 0.0000013828492 orOne chance in 723,144.64.

Match 4 out of 5 white balls but not match the Powerball (Payout = $100)The number of ways 4 of the 5 winning numbers on your lottery ticket can match the 5 white balls is COMBIN(5,4). The number of ways the losing white number on your ticket can match any of the 54 losing numbers is COMBIN54,1) = 54. The number of ways your Powerball number can miss matching the single Powerball number is:

COMBIN(38,1) = 38. The product of these is the number of ways you can win this configuration: COMBIN(5,4) xCOMBIN(54,1) x COMBIN(38,1) = 10,260. The probability of success is thus: 10,260/195,249,054 = 0.000052548or One chance in 19,030.12.

Match 3 out of 5 white balls and match the Powerball (Payout = $100)The number of ways 3 of the 5 winning numbers on your lottery ticket can match the 5 white balls is COMBIN(5,3)0. The number of ways the 2 losing white numbers on your ticket can match any of the 54 losing white numbers is

COMBIN(54,2) = 1,431. The number of ways your Powerball number can match the single Powerball number is:COMBIN(1,1) = 1. The product of these is the number of ways you can win this configuration: COMBIN(5,3) xCOMBIN(54,2) x COMBIN(1,1) = 14,310. The probability of success is thus: 14,310/195,249,054 = 0.000073291 oOne chance in 13,644.24.

Match 3 out of 5 white balls but not match the Powerball (Payout = $7)The number of ways 3 of the 5 winning numbers on your lottery ticket can match the 5 white balls is COMBIN(5,3)0. The number of ways the 2 losing white numbers on your ticket can match any of the 54 losing numbers is

COMBIN(54,2) = 1,431. The number of ways your Powerball number can miss matching the single Powerball nums: COMBIN(38,1) = 38. The product of these is the number of ways you can win this configuration: COMBIN(5,3

COMBIN(54,2) x COMBIN(38,1) = 543,780. The probability of success is thus: 543,780/195,249,054 = 0.0027850or One chance in 359.06.

Match 2 out of 5 white balls and match the Powerball (Payout = $7)The number of ways 2 of the 5 winning numbers on your lottery ticket can match the 5 white balls is COMBIN(5,2)0. The number of ways the 3 losing white numbers on your ticket can match any of the 54 losing white numbers is

COMBIN(54,3) = 24,804. The number of ways your Powerball number can match the single Powerball number is:COMBIN(1,1) = 1. The product of these is the number of ways you can win this configuration: COMBIN(5,2) xCOMBIN(54,3) x COMBIN(1,1) = 248,040. The probability of success is thus: 248,040/195,249,054 = 0.00127037One chance in 787.17.

Match 1 out of 5 white balls and match the Powerball (Payout = $4)The number of ways 1 of the 5 winning numbers on your lottery ticket can match the 5 white balls is COMBIN(5,1). The number of ways the 4 losing white numbers on your ticket can match any of the 54 losing white numbers is

COMBIN(54,4) = 316,251. The number of ways your Powerball number can match the single Powerball number isCOMBIN(1,1) = 1. The product of these is the number of ways you can win this configuration: COMBIN(5,1) xCOMBIN(54,4) x COMBIN(1,1) = 1,581,255. The probability of success is thus: 1,581,255/195,249,054 =0.008098656 or One chance in 123.48.




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Match 0 out of 5 white balls and match the Powerball (Payout = $3)The number of ways 0 of the 5 winning numbers on your lottery ticket can match the 5 white balls is COMBIN(5,0). The number of ways the 5 losing white numbers on your ticket can match any of the 54 losing white numbers is

COMBIN(54,5) = 3,162,510. The number of ways your Powerball number can match the single Powerball numberCOMBIN(1,1) = 1. The product of these is the number of ways you can win this configuration: COMBIN(5,0) xCOMBIN(54,5) x COMBIN(1,1) = 3,162,510. The probability of success is thus: 3,162,510/195,249,054 =0.016197313 or One chance in 61.74.

Probability of winning somethingf we add all the ways you can win something we get: + 38 + 270 + 10,260 + 14,310 + 543,780 + 248,040 + 1,581,255 + 3,162,510 = 5,560,464 different ways of winninomething. If we divide this number by 195,249,054, we get .028478827 as a probability of winning something. 1

divided by 0.028478827 yields One chance in 35.11 of winning something.

CorollaryYou can get a close estimate for the number of tickets that were in play for any given game by multiplying the

nnounced number of winners by the above 35.11. Thus, if the lottery officials proclaim that a given lottery drawihad 2 million winners, then there were about 2,000,000 x 35.11 ~= 70,220,000 tickets purchased that did not win ackpot. Alternately, there were about 70,220,000 - 2,000,000 ~= 68,220,000 tickets that did not win anything.

Probability of multiple winning tickets (multiple winners) given N tickets in play

Note: All calculations assume that the numbers on any given ticket are picked randomly. In practice, many people pnumbers based on family birthdays, etc., and thus many tickets will have a preponderance of low numbers. As aonsequence, the probabilities of a single Jackpot winner will be somewhat lower and the probabilities of no winner

multiple winners will tend to be slightly higher than the numbers shown below. Also if the numbers picked in thedrawing are clustered at the high end of the 1-59 range, there will tend to be relatively less partial match winners.The reverse will hold true if the drawing numbers cluster in the low end of the number range.)

The above chart shows the probabilities of No Winners, One Winner, and Two or more Winners for various

numbers of tickets in play.




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Each entry in the following table shows the probability of "K" tickets holding the same winning Jackpot combinatgiven that "N" tickets are in play for a given Powerball game. It is assumed that the number selections on each tickere picked randomly. For example, if 100,000,000 tickets are in play for a Powerball game, then there is a 0.0786

probability that exactly two of these tickets will have the same winning combination.

(Note: You can get a rough estimate of the number of tickets in play as follows. If the preceding Powerball game hno Jackpot winner, multiply the dollar increase in the Jackpot from the preceding game to the current game by 3. (Uhe cash payout amount). For example, if the preceding game had a cash payout amount of $50,000,000 and the curr

game has a cash payout amount of $70,000,000, then there are about 3 x (70,000,000 - 50,000,000) = 60,000,000ickets in play for the current game. A history of these past jackpot amounts (subtract about 50 % from the statedackpot amount to get the cash payout) can be seen at:

http://www.lottostrategies.com/script/jackpot_history/draw_date/101)

N Number K

f tickets Number of tickets holding the Jackpot combination

n play 0 1 2 3 4 5 6

---------------------------------------------------------------------

0,000,000 0.9026 0.0925 0.0047 0.0002 0.0000 0.0000 0.0000

0,000,000 0.8148 0.1669 0.0171 0.0012 0.0001 0.0000 0.0000

0,000,000 0.7354 0.2260 0.0347 0.0036 0.0003 0.0000 0.0000

0,000,000 0.6638 0.2720 0.0557 0.0076 0.0008 0.0001 0.0000

00,000,000 0.5992 0.3069 0.0786 0.0134 0.0017 0.0002 0.000020,000,000 0.5409 0.3324 0.1021 0.0209 0.0032 0.0004 0.0000

40,000,000 0.4882 0.3501 0.1255 0.0300 0.0054 0.0008 0.0001

60,000,000 0.4407 0.3611 0.1480 0.0404 0.0083 0.0014 0.0002

80,000,000 0.3978 0.3667 0.1690 0.0519 0.0120 0.0022 0.0003

00,000,000 0.3590 0.3678 0.1884 0.0643 0.0165 0.0034 0.0006

20,000,000 0.3241 0.3652 0.2057 0.0773 0.0218 0.0049 0.0009

40,000,000 0.2925 0.3596 0.2210 0.0905 0.0278 0.0068 0.0014

60,000,000 0.2640 0.3516 0.2341 0.1039 0.0346 0.0092 0.0020

80,000,000 0.2383 0.3418 0.2451 0.1172 0.0420 0.0120 0.0029

00,000,000 0.2151 0.3306 0.2539 0.1301 0.0500 0.0154 0.0039

20,000,000 0.1942 0.3183 0.2608 0.1425 0.0584 0.0191 0.0052

40,000,000 0.1753 0.3052 0.2658 0.1543 0.0672 0.0234 0.0068

60,000,000 0.1582 0.2917 0.2689 0.1653 0.0762 0.0281 0.0086

80,000,000 0.1428 0.2779 0.2705 0.1755 0.0854 0.0332 0.010800,000,000 0.1289 0.2641 0.2705 0.1847 0.0946 0.0388 0.0132

Any entry in the table can be calculated using the following equation:

Prob. = COMBIN(N,K) x (Pwin^K) x (Pnotwin^(N-K))

Where:N = Number of tickets in playK = Number of tickets holding the Jackpot combinationPwin = Probability that a random ticket will win ( = 1 / 195,249,054 = 0.0000000051)Pnotwin = (1.0 - Pwin) = 0.9999999949COMBIN(N,K) = number of ways to select K items from a group of N itemsx = multiply terms = raise to power (e.g. 2^3 = 8 )

Sample Calculation to Find the Expected Shared Jackpot AmountWhen a Large Number of Tickets are in Play

For this example we will assume the cash value of the Jackpot is $120,000,000 and there are 140,000,000 tickets inplay for the current game. Probability values are from the 140,000,000 row above.

Number of Jackpot paid Contribution




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winners Probability to each winner (Col 2 x Col 3)

-------------------------------------------------------------

.4882 0 0

.3501 120,000,000 42,012,000

.1255 60,000,000 7,530,000

.0300 40,000,000 1,200,000

.0054 30,000,000 162,000

.0008 24,000,000 19,200

.0001 20,000,000 2,000

otal 50,925,200

This total then has to be divided by 1 - .4882 = .5118 to give a weighted Jackpot amount of 50,925,200 / .5118 ~=$99,502,149 which would be used as the payout amount figure used in the Return on Investment section below.

These calculations can be used to form an index showing how much the quoted amount of the Jackpot should beeduced to allow for the expected number of co-winners. In the table below, for any given number of tickets in play,

quoted Jackpot should be multiplied by the value in the next column to produce the true expected value of a winningicket. For example, if there are 200,000,000 tickets in play for a quoted $300,000,000 Jackpot, then the expected vaor the Jackpot becomes $300,000,000 x 0.7618 = $228,540,000 to adjust for the possibility that a winning ticket w

have to split the Jackpot with some other winning ticket.

Number of Mult. Jackpot by Number of Mult. Jackpot by

Tickets this ratio for Tickets this ratio for

in play possible sharing in play possible sharing

0 1.0000 200,000,000 0.7618

20,000,000 0.9745 220,000,000 0.7403

40,000,000 0.9494 240,000,000 0.7192

60,000,000 0.9246 260,000,000 0.6986

80,000,000 0.9001 280,000,000 0.6785

00,000,000 0.8760 300,000,000 0.6588

20,000,000 0.8524 320,000,000 0.6397

40,000,000 0.8291 340,000,000 0.6210

60,000,000 0.8062 360,000,000 0.6027

80,000,000 0.7838 380,000,000 0.5850

00,000,000 0.7618 400,000,000 0.5677

Power Play Multiplier

The Powerball game includes an optional Multiplier. If you spend an extra $1 for the multiplier, then the low orpayouts are multiplied by the Multiplier. The payout for match 5 white balls but not the powerball becomes$1,000,000 no matter what the multiplier is. Finally, there is no change for the Jackpot amount. The probability of a, 4, or 5 for the multiplier is 0.25 each.

The net effect of the multiplier is found by multiplying the probability of each outcome by the resulting digit,

dding the results together, and then subtracting 1.00. (1.00 is subtracted as you would get this payout even if you juplayed the regular game.) Thus we can calculate the weighted multiplier amount as follows:Weighted Multiplier = 0.25 x 2 + 0.25 x 3 + 0.25 x 4 + 0.25 x 5 1.00 = 2.5We will use this result in the Return on Investment section.

Return on Investment

Finally, it is interesting to calculate what the long term expected return is for each $1.00 lottery ticket that you buyWe will also calculate the return on the optional Power Play multiplier.




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The first task is to construct a table where each row lists the winning combination, the payout, the probability of thpayout, and the contribution to the expected return (Equals payout times probability.) The probabilities are the sameones we derived earlier. A $64,000,000 cash payout (decline the annuity) is assumed for the Jackpot. (Could be yourportion of a shared Jackpot.)ombination Payout Probability Contribution

------------------------------------------------------

White + PB $64,000,000 5.12166E-09 $0.3278

White No PB 200,000 1.94623E-07 0.0389

White + PB 10,000 1.38285E-06 0.0138

White No PB 100 5.25483E-05 0.0053

White + PB 100 7.32910E-05 0.0073

White No PB 7 0.002785058 0.0195

White + PB 7 0.001270377 0.0089

White + PB 4 0.008098656 0.0324

B 3 0.016197313 0.0486

otal 0.028478827 0.5025

otal for last 7 rows 0.1358

Thus, for each $1.00 that you spend for Powerball tickets, you can expect to get back about $0.50. Of course you go pay taxes on any large payout, so your net return is even less.

Next, we can calculate the expected return if you pay another $1.00 for the Power Play Multiplier. Here we use $0.1358 from the last 7 rows as the multiplier is used for all low-order payouts except for matching 5 white balls. Wlso have to add in the expected return for the $1,000,000 for matching all 5 white balls. The expected return for the

$1,000,000 payout is the incremental increase that you get for the multiplier option ($1,000,000 - $200,000 =$800,000) times its probability which evaluates to $800,000 times 1.94623E-07 = $0.1557. When we multiply the$0.1358 by the Weighted Multiplier of 2.5 that we calculated earlier and then add the $0.1557, we get: 0.1358 x 20.1557 = $0.4952. Thus, for each $1.00 that you pay for the Power Play Multiplier, your long run expected returno get back a little under 50 cents.

Expected after tax return on your $1.00 ticket investment when a large Jackpot is in play

While the above calculation represents an average Powerball game, we might ask what the expected after tax returon your investment might be if a large Jackpot exists. The following analysis assumes the annuity value of the Jackps $400 million and there are 200 million tickets in play. The cash value for any Jackpot is about one-half the annuit

value which brings the real value down to $200,000,000. We will also ignore any carryover bonus. All prizes of$10,000 and above are reduced 40% to allow for federal and state taxes. Dont forget that a large prize will throw yonto a top tax bracket.

First, lets calculate the effective Jackpot payout based on 200 million tickets in play. (Please see the Shared JackAmount When a Large Number of Tickets are in Play section for the calculation method, but we will use the 200million row.) Thus:0.3678 x 200000000 + 0.1884 x 200000000/2 + 0.0643 x 200000000/3 + 0.0165 x 200000000/4 + 0.0034 x

200000000/5 + 0.0006 x 200000000/6) / (1 - 0.3590) = $152,367,655. This is the before taxes, effective cash Jackpomount, adjusted for the possibility that you will have to share the Jackpot if you win. Then subtract 40% for taxes

which will leave an after tax Jackpot of $91,420,593. Then multiply by the probability that you will win this Jackpowhich yields: 91,420,593 x 5.12166E-09 = $0.4682 expected after tax return from the Jackpot.

Earlier we calculated a before tax expected return of $0.0389+ for Match 5 but not the powerball. If we subtract40% for taxes we get an after tax expected return of $0.0234. Similarly we previously found a before tax return of$0.0138 for 4 White + PB. Subtracting 40% for taxes leaves an after tax expected return of $0.0083. For all smallprizes we assume that you dont report your winnings. Thus we just add in the (0.0053 + 0.0073 + 0.0195 + 0.0089 0.0324 + 0.0486) = 0.1220

Finally, to get the expected after tax return on your $1.00 ticket purchase, we just find the sum of all the above par




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esults. $0.4682 + 0.0234 + 0.0083 + 0.1220 ~= $0.6218. Thus, even for a huge Jackpot with a quoted $400 millionpayout, your after tax expected return is only about $0.62 for every $1.00 ticket that you buy.

2nd Thoughts

Government statistics show there are about 1.7 automobile caused fatalities for every 100,000,000 vehicle-miles. I

you drive one mile to the store to buy your lottery ticket and then return home, you have driven two miles. Thus theprobability that you will join this statistical group is 2 x 1.7 / 100,000,000 = 0.000000034. This can also be stated asOne in 29,411,765-. Thus, if you drive to the store to buy your Powerball ticket, your chance of being killed (or

killing someone else) is nearly 7 times greater than the chance that you will win the Powerball Jackpot.

3rd Thoughts

A lottery is a Zero-sum game. What one group of participants gains in cash, the other group of participants musose. If we made a list of all the participants in a lottery, it might include:

) Federal Government (Lottery winnings are taxable)2) State Governments (Again lottery winnings are taxable)) State Governments (Direct share of lottery ticket sales)

4) Merchants that sell tickets (Paid by the lottery organizers)) Lottery companies (Hint: They are not doing all this for free)

6) Advertisers and promoters (Paid by the lottery companies)7) Lottery ticket buyers (Buy lottery tickets and receive payouts)

The winners in the above list are:) Federal Government

2) State Government (Taxes)

) State Government (Direct share)4) Merchants that sell tickets) Lottery companies

6) Advertisers and promoters

And the losers are:(Mathematically challenged and proud of it)

Also please see the related calculations for Mega Millions

Note about Googles/Yahoos search engines

For reasons unknown and for which Yahoo refuses to disclose, this entire website has been blacklisted/banned byYahoos search engine. Other websites have suffered a similar fate. If you are trying to find information via Googleearch engine vs. Yahoos search engine, you should understand that Yahoos results may not include the informatiohat you are seeking.

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