Applied Mathematics
Powerball Odds
How to Calculate the Probabilities for the
Powerball 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-State Lottery gives a concise table at their web site.
We will give the same information here, but also show you how these
odds are calculated.
Game Rules
The numbers picked for the prizes consist of 5 white balls picked at random from a drum that holds 59 balls numbered from 1 to 59. The Powerball number is a single ball that is picked from a second drum that has 39 numbers ranging from 1 to 39. If the results of these random number selections match one of the winning combinations on your lottery 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 is that 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 turn will lead to somewhat higher average Jackpots which appears to be Powerball’s answer to the very large Jackpots which sometimes develop in the Mega Millions lottery. For any given number of tickets in play, the probability that there will be multiple winners is significantly decreased.
In any combinatorics problem where all possible outcomes are equally likely, the probability of a successful outcome is determined by finding the number of successful combinations, and then dividing by the total number of all combinations. There are nine possible configurations that will win something in the Powerball Lottery. For each of these, the probability of winning equals the number of winning combinations for that particular configuration divided by the total number of ways the Powerball numbers can be picked.
Powerball Total Combinations
Since the total number of combinations for Powerball numbers is used in all the calculations, we will calculate it first. The number of ways 5 numbers can be randomly selected from a field of 59 is: COMBIN(59,5) = 5,006,386. (See the math 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 Powerball number. The total number of ways to pick the 6 numbers is the product of these. Thus, the total number of equally likely Powerball combinations is 5,006,386 x 39 = 195,249,054. We will use this number for each of the following calculations.
Jackpot 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. 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 the Jackpot: COMBIN(5,5) x COMBIN(1,1) = 1. The probability of success is thus: 1/195,249,054 = 0.000000005121663739+. If you express this as “One chance in ???”, you just divide “1” by the 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. The number of ways your Powerball number can match any of the 38 losing Powerball numbers is: COMBIN(38,1) = 38. (Pick 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) = 5. The number of ways the losing white number on your ticket can match any of the 54 losing white numbers is COMBIN(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) x COMBIN(54,1) x COMBIN(1,1) = 270. The probability of success is thus: 270/195,249,054 = 0.0000013828492 or “One 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) = 5. The number of ways the losing white number on your ticket can match any of the 54 losing numbers is COMBIN(54,1) = 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) x COMBIN(54,1) x COMBIN(38,1) = 10,260. The probability of success is thus: 10,260/195,249,054 = 0.00005254827 or “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) = 10. 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) x COMBIN(54,2) x COMBIN(1,1) = 14,310. The probability of success is thus: 14,310/195,249,054 = 0.000073291 or “One 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) = 10. 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 number is: COMBIN(38,1) = 38. The product of these is the number of ways you can win this configuration: COMBIN(5,3) x COMBIN(54,2) x COMBIN(38,1) = 543,780. The probability of success is thus: 543,780/195,249,054 = 0.002785058 or “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) = 10. 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) x COMBIN(50,3) x COMBIN(1,1) = 248,040. The probability of success is thus: 248,040/195,249,054 = 0.001270377 or “One 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) = 5. 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 is: COMBIN(1,1) = 1. The product of these is the number of ways you can win this configuration: COMBIN(5,1) x COMBIN(50,4) 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”.
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) = 1. 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 number is: COMBIN(1,1) = 1. The product of these is the number of ways you can win this configuration: COMBIN(5,0) x COMBIN(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 something
If we add all the ways you can win something we get:
1 + 38 + 270 + 10,260 + 14,310 + 543,780 + 248,040 + 1,581,255 + 3,162,510 = 5,560,464 different ways of winning something. 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.
Corollary
You can get a close estimate for the number of tickets that were in play for any given game by multiplying the announced number of “winners” by the above 35.11. Thus, if the lottery officials proclaim that a given lottery drawing had 2 million “winners”, then there were about 2,000,000 x 35.11 ~= 70,220,000 tickets purchased that did not win the Jackpot. Alternately, there were about 70,220,000 - 2,000,000 ~= 68,220,000 tickets that did not win anything.
(Note: All calculations assume that the numbers on any given ticket are picked randomly. In practice, many people pick numbers based on family birthdays, etc., and thus many tickets will have a preponderance of low numbers. As a consequence, the probabilities of a single Jackpot winner will be somewhat lower and the probabilities of no winner or multiple winners will tend to be slightly higher than the numbers shown below. Also if the numbers picked in the drawing 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.
Each entry in the following table shows the probability of "K" tickets holding the same winning Jackpot combination given that "N" tickets are in play for a given Powerball game. It is assumed that the number selections on each ticket are 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 had no Jackpot winner, multiply the dollar increase in the Jackpot from the preceding game to the current game by 3. (Use the cash payout amount). For example, if the preceding game had a cash payout amount of $50,000,000 and the current game has a cash payout amount of $70,000,000, then there are about 3 x (70,000,000 - 50,000,000) = 60,000,000 tickets in play for the current game. A history of these past jackpot amounts (subtract about 50 % from the stated jackpot amount to get the cash payout) can be seen at: http://www.lottostrategies.com/script/jackpot_history/draw_date/101)
“N” Number “K”
of tickets Number of tickets holding the Jackpot combination
in play 0 1 2 3 4 5 6
----------------------------------------------------------------------
20,000,000 0.9026 0.0925 0.0047 0.0002 0.0000 0.0000 0.0000
40,000,000 0.8148 0.1669 0.0171 0.0012 0.0001 0.0000 0.0000
60,000,000 0.7354 0.2260 0.0347 0.0036 0.0003 0.0000 0.0000
80,000,000 0.6638 0.2720 0.0557 0.0076 0.0008 0.0001 0.0000
100,000,000 0.5992 0.3069 0.0786 0.0134 0.0017 0.0002 0.0000
120,000,000 0.5409 0.3324 0.1021 0.0209 0.0032 0.0004 0.0000
140,000,000 0.4882 0.3501 0.1255 0.0300 0.0054 0.0008 0.0001
160,000,000 0.4407 0.3611 0.1480 0.0404 0.0083 0.0014 0.0002
180,000,000 0.3978 0.3667 0.1690 0.0519 0.0120 0.0022 0.0003
200,000,000 0.3590 0.3678 0.1884 0.0643 0.0165 0.0034 0.0006
220,000,000 0.3241 0.3652 0.2057 0.0773 0.0218 0.0049 0.0009
240,000,000 0.2925 0.3596 0.2210 0.0905 0.0278 0.0068 0.0014
260,000,000 0.2640 0.3516 0.2341 0.1039 0.0346 0.0092 0.0020
280,000,000 0.2383 0.3418 0.2451 0.1172 0.0420 0.0120 0.0029
300,000,000 0.2151 0.3306 0.2539 0.1301 0.0500 0.0154 0.0039
320,000,000 0.1942 0.3183 0.2608 0.1425 0.0584 0.0191 0.0052
340,000,000 0.1753 0.3052 0.2658 0.1543 0.0672 0.0234 0.0068
360,000,000 0.1582 0.2917 0.2689 0.1653 0.0762 0.0281 0.0086
380,000,000 0.1428 0.2779 0.2705 0.1755 0.0854 0.0332 0.0108
400,000,000 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 play
K = Number of tickets holding the Jackpot combination
Pwin = Probability that a random ticket will win ( = 1 / 195,249,054 = 0.0000000051)
Pnotwin = (1.0 - Pwin) = 0.9999999949
COMBIN(N,K) = number of ways to select K items from a group of N items
x = multiply terms
^ = raise to power (e.g. 2^3 = 8 )
Sample Calculation to Find the Expected Shared Jackpot Amount
When 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 in play for the current game. Probability values are from the “140,000,000” row above.
Number of Jackpot paid Contribution
winners Probability to each winner (Col 2 x Col 3)
--------------------------------------------------------------
0 .4882 0 0
1 .3501 120,000,000 42,012,000
2 .1255 60,000,000 7,530,000
3 .0300 40,000,000 1,200,000
4 .0054 30,000,000 162,000
5 .0008 24,000,000 19,200
6 .0001 20,000,000 2,000
Total 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 be reduced to allow for the expected number of co-winners. In the table below, for any given number of tickets in play, the quoted Jackpot should be multiplied by the value in the next column to produce the true expected value of a winning ticket. For example, if there are 200,000,000 tickets in play for a quoted $300,000,000 Jackpot, then the expected value for the Jackpot becomes $300,000,000 x 0.7618 = $228,540,000 to adjust for the possibility that a winning ticket will 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
100,000,000 0.8760 300,000,000 0.6588
120,000,000 0.8524 320,000,000 0.6397
140,000,000 0.8291 340,000,000 0.6210
160,000,000 0.8062 360,000,000 0.6027
180,000,000 0.7838 380,000,000 0.5850
200,000,000 0.7618 400,000,000 0.5677
The Powerball game includes an optional “Multiplier”. If you spend an extra $1 for the multiplier, then the low order payouts 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 2, 3, 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, adding the results together, and then subtracting 1.00. (1.00 is subtracted as you would get this payout even if you just played 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 = 2.5
We will use this result in the “Return on Investment” section.
Finally, it is interesting to calculate what the long term expected return is for each $1.00 lottery ticket that you buy. We will also calculate the return on the optional Power Play multiplier.
The first task is to construct a table where each row lists the winning combination, the payout, the probability of this payout, and the contribution to the expected return (Equals payout times probability.) The probabilities are the same ones we derived earlier. A $64,000,000 cash payout (decline the annuity) is assumed for the Jackpot. (Could be your portion of a shared Jackpot.)
Combination Payout Probability Contribution
-------------------------------------------------------
5 White + PB $64,000,000 5.12166E-09 $0.3278
5 White No PB 200,000 1.94623E-07 0.0389
4 White + PB 10,000 1.38285E-06 0.0138
4 White No PB 100 5.25483E-05 0.0053
3 White + PB 100 7.32910E-05 0.0073
3 White No PB 7 0.002785058 0.0195
2 White + PB 7 0.001270377 0.0089
1 White + PB 4 0.008098656 0.0324
PB 3 0.016197313 0.0486
Total 0.028478827 0.5025
Total 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 get to 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 the $0.1358 from the last 7 rows as the multiplier is used for all low-order payouts except for matching 5 white balls. We also 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 this number times its probability which evaluates to $1,000,000 times 1.94623E-07 = $0.1946. When we multiply the $0.1358 by the “Weighted Multiplier” of 2.5 that we calculated earlier and then add the $0.1946, we get: 0.1358 x 2.5 + 0.1946 = $0.5341. Thus, for each $1.00 that you pay for the “Power Play Multiplier”, your long run expected return is to get back about 53.4 cents.
While the above calculation represents an average Powerball game, we might ask what the expected after tax return on your investment might be if a large Jackpot exists. The following analysis assumes the annuity value of the Jackpot is $400 million and there are 200 million tickets in play. The cash value for any Jackpot is about one-half the annuity 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. Don’t forget that a large prize will throw you into a top tax bracket.
First, let’s calculate the effective Jackpot payout based on 200 million tickets in play. (Please see the “Shared Jackpot Amount When a Large Number of Tickets are in Play” section for the calculation method, but we will use the 200 million 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 Jackpot amount, 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 Jackpot which 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 subtract 40% 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 smaller prizes we assume that you don’t 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 partial results. $0.4682 + 0.0234 + 0.0083 + 0.1220 ~= $0.6218. Thus, even for a huge Jackpot with a quoted $400 million payout, your after tax expected return is only about $0.62 for every $1.00 ticket that you buy.
Government statistics show there are about 1.7 automobile caused fatalities for every 100,000,000 vehicle-miles. If you drive one mile to the store to buy your lottery ticket and then return home, you have driven two miles. Thus the probability that you will join this statistical group is 2 x 1.7 / 100,000,000 = 0.000000034. This can also be stated as “One 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.
A lottery is a “Zero-sum game”. What one group of participants gains in cash, the other group of participants must lose. If we made a list of all the participants in a lottery, it might include:
1) Federal Government (Lottery winnings are taxable)
2) State Governments (Again lottery winnings are taxable)
3) State Governments (Direct share of lottery ticket sales)
4) Merchants that sell tickets (Paid by the lottery organizers)
5) 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:
1) Federal Government
2) State Government (Taxes)
3) State Government (Direct share)
4) Merchants that sell tickets
5) 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
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Game Rules
The numbers picked for the prizes consist of 5 white balls picked at random from a drum that holds 59 balls numbered from 1 to 59. The Powerball number is a single ball that is picked from a second drum that has 39 numbers ranging from 1 to 39. If the results of these random number selections match one of the winning combinations on your lottery 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 is that 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 turn will lead to somewhat higher average Jackpots which appears to be Powerball’s answer to the very large Jackpots which sometimes develop in the Mega Millions lottery. For any given number of tickets in play, the probability that there will be multiple winners is significantly decreased.
In any combinatorics problem where all possible outcomes are equally likely, the probability of a successful outcome is determined by finding the number of successful combinations, and then dividing by the total number of all combinations. There are nine possible configurations that will win something in the Powerball Lottery. For each of these, the probability of winning equals the number of winning combinations for that particular configuration divided by the total number of ways the Powerball numbers can be picked.
Powerball Total Combinations
Since the total number of combinations for Powerball numbers is used in all the calculations, we will calculate it first. The number of ways 5 numbers can be randomly selected from a field of 59 is: COMBIN(59,5) = 5,006,386. (See the math 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 Powerball number. The total number of ways to pick the 6 numbers is the product of these. Thus, the total number of equally likely Powerball combinations is 5,006,386 x 39 = 195,249,054. We will use this number for each of the following calculations.
Jackpot 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. 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 the Jackpot: COMBIN(5,5) x COMBIN(1,1) = 1. The probability of success is thus: 1/195,249,054 = 0.000000005121663739+. If you express this as “One chance in ???”, you just divide “1” by the 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. The number of ways your Powerball number can match any of the 38 losing Powerball numbers is: COMBIN(38,1) = 38. (Pick 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) = 5. The number of ways the losing white number on your ticket can match any of the 54 losing white numbers is COMBIN(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) x COMBIN(54,1) x COMBIN(1,1) = 270. The probability of success is thus: 270/195,249,054 = 0.0000013828492 or “One 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) = 5. The number of ways the losing white number on your ticket can match any of the 54 losing numbers is COMBIN(54,1) = 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) x COMBIN(54,1) x COMBIN(38,1) = 10,260. The probability of success is thus: 10,260/195,249,054 = 0.00005254827 or “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) = 10. 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) x COMBIN(54,2) x COMBIN(1,1) = 14,310. The probability of success is thus: 14,310/195,249,054 = 0.000073291 or “One 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) = 10. 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 number is: COMBIN(38,1) = 38. The product of these is the number of ways you can win this configuration: COMBIN(5,3) x COMBIN(54,2) x COMBIN(38,1) = 543,780. The probability of success is thus: 543,780/195,249,054 = 0.002785058 or “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) = 10. 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) x COMBIN(50,3) x COMBIN(1,1) = 248,040. The probability of success is thus: 248,040/195,249,054 = 0.001270377 or “One 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) = 5. 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 is: COMBIN(1,1) = 1. The product of these is the number of ways you can win this configuration: COMBIN(5,1) x COMBIN(50,4) 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”.
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) = 1. 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 number is: COMBIN(1,1) = 1. The product of these is the number of ways you can win this configuration: COMBIN(5,0) x COMBIN(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 something
If we add all the ways you can win something we get:
1 + 38 + 270 + 10,260 + 14,310 + 543,780 + 248,040 + 1,581,255 + 3,162,510 = 5,560,464 different ways of winning something. 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.
Corollary
You can get a close estimate for the number of tickets that were in play for any given game by multiplying the announced number of “winners” by the above 35.11. Thus, if the lottery officials proclaim that a given lottery drawing had 2 million “winners”, then there were about 2,000,000 x 35.11 ~= 70,220,000 tickets purchased that did not win the Jackpot. 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 pick numbers based on family birthdays, etc., and thus many tickets will have a preponderance of low numbers. As a consequence, the probabilities of a single Jackpot winner will be somewhat lower and the probabilities of no winner or multiple winners will tend to be slightly higher than the numbers shown below. Also if the numbers picked in the drawing 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.
Each entry in the following table shows the probability of "K" tickets holding the same winning Jackpot combination given that "N" tickets are in play for a given Powerball game. It is assumed that the number selections on each ticket are 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 had no Jackpot winner, multiply the dollar increase in the Jackpot from the preceding game to the current game by 3. (Use the cash payout amount). For example, if the preceding game had a cash payout amount of $50,000,000 and the current game has a cash payout amount of $70,000,000, then there are about 3 x (70,000,000 - 50,000,000) = 60,000,000 tickets in play for the current game. A history of these past jackpot amounts (subtract about 50 % from the stated jackpot amount to get the cash payout) can be seen at: http://www.lottostrategies.com/script/jackpot_history/draw_date/101)
“N” Number “K”
of tickets Number of tickets holding the Jackpot combination
in play 0 1 2 3 4 5 6
----------------------------------------------------------------------
20,000,000 0.9026 0.0925 0.0047 0.0002 0.0000 0.0000 0.0000
40,000,000 0.8148 0.1669 0.0171 0.0012 0.0001 0.0000 0.0000
60,000,000 0.7354 0.2260 0.0347 0.0036 0.0003 0.0000 0.0000
80,000,000 0.6638 0.2720 0.0557 0.0076 0.0008 0.0001 0.0000
100,000,000 0.5992 0.3069 0.0786 0.0134 0.0017 0.0002 0.0000
120,000,000 0.5409 0.3324 0.1021 0.0209 0.0032 0.0004 0.0000
140,000,000 0.4882 0.3501 0.1255 0.0300 0.0054 0.0008 0.0001
160,000,000 0.4407 0.3611 0.1480 0.0404 0.0083 0.0014 0.0002
180,000,000 0.3978 0.3667 0.1690 0.0519 0.0120 0.0022 0.0003
200,000,000 0.3590 0.3678 0.1884 0.0643 0.0165 0.0034 0.0006
220,000,000 0.3241 0.3652 0.2057 0.0773 0.0218 0.0049 0.0009
240,000,000 0.2925 0.3596 0.2210 0.0905 0.0278 0.0068 0.0014
260,000,000 0.2640 0.3516 0.2341 0.1039 0.0346 0.0092 0.0020
280,000,000 0.2383 0.3418 0.2451 0.1172 0.0420 0.0120 0.0029
300,000,000 0.2151 0.3306 0.2539 0.1301 0.0500 0.0154 0.0039
320,000,000 0.1942 0.3183 0.2608 0.1425 0.0584 0.0191 0.0052
340,000,000 0.1753 0.3052 0.2658 0.1543 0.0672 0.0234 0.0068
360,000,000 0.1582 0.2917 0.2689 0.1653 0.0762 0.0281 0.0086
380,000,000 0.1428 0.2779 0.2705 0.1755 0.0854 0.0332 0.0108
400,000,000 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 play
K = Number of tickets holding the Jackpot combination
Pwin = Probability that a random ticket will win ( = 1 / 195,249,054 = 0.0000000051)
Pnotwin = (1.0 - Pwin) = 0.9999999949
COMBIN(N,K) = number of ways to select K items from a group of N items
x = multiply terms
^ = raise to power (e.g. 2^3 = 8 )
Sample Calculation to Find the Expected Shared Jackpot Amount
When 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 in play for the current game. Probability values are from the “140,000,000” row above.
Number of Jackpot paid Contribution
winners Probability to each winner (Col 2 x Col 3)
--------------------------------------------------------------
0 .4882 0 0
1 .3501 120,000,000 42,012,000
2 .1255 60,000,000 7,530,000
3 .0300 40,000,000 1,200,000
4 .0054 30,000,000 162,000
5 .0008 24,000,000 19,200
6 .0001 20,000,000 2,000
Total 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 be reduced to allow for the expected number of co-winners. In the table below, for any given number of tickets in play, the quoted Jackpot should be multiplied by the value in the next column to produce the true expected value of a winning ticket. For example, if there are 200,000,000 tickets in play for a quoted $300,000,000 Jackpot, then the expected value for the Jackpot becomes $300,000,000 x 0.7618 = $228,540,000 to adjust for the possibility that a winning ticket will 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
100,000,000 0.8760 300,000,000 0.6588
120,000,000 0.8524 320,000,000 0.6397
140,000,000 0.8291 340,000,000 0.6210
160,000,000 0.8062 360,000,000 0.6027
180,000,000 0.7838 380,000,000 0.5850
200,000,000 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 order payouts 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 2, 3, 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, adding the results together, and then subtracting 1.00. (1.00 is subtracted as you would get this payout even if you just played 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 = 2.5
We 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 buy. We will also calculate the return on the optional Power Play multiplier.
The first task is to construct a table where each row lists the winning combination, the payout, the probability of this payout, and the contribution to the expected return (Equals payout times probability.) The probabilities are the same ones we derived earlier. A $64,000,000 cash payout (decline the annuity) is assumed for the Jackpot. (Could be your portion of a shared Jackpot.)
Combination Payout Probability Contribution
-------------------------------------------------------
5 White + PB $64,000,000 5.12166E-09 $0.3278
5 White No PB 200,000 1.94623E-07 0.0389
4 White + PB 10,000 1.38285E-06 0.0138
4 White No PB 100 5.25483E-05 0.0053
3 White + PB 100 7.32910E-05 0.0073
3 White No PB 7 0.002785058 0.0195
2 White + PB 7 0.001270377 0.0089
1 White + PB 4 0.008098656 0.0324
PB 3 0.016197313 0.0486
Total 0.028478827 0.5025
Total 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 get to 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 the $0.1358 from the last 7 rows as the multiplier is used for all low-order payouts except for matching 5 white balls. We also 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 this number times its probability which evaluates to $1,000,000 times 1.94623E-07 = $0.1946. When we multiply the $0.1358 by the “Weighted Multiplier” of 2.5 that we calculated earlier and then add the $0.1946, we get: 0.1358 x 2.5 + 0.1946 = $0.5341. Thus, for each $1.00 that you pay for the “Power Play Multiplier”, your long run expected return is to get back about 53.4 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 return on your investment might be if a large Jackpot exists. The following analysis assumes the annuity value of the Jackpot is $400 million and there are 200 million tickets in play. The cash value for any Jackpot is about one-half the annuity 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. Don’t forget that a large prize will throw you into a top tax bracket.
First, let’s calculate the effective Jackpot payout based on 200 million tickets in play. (Please see the “Shared Jackpot Amount When a Large Number of Tickets are in Play” section for the calculation method, but we will use the 200 million 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 Jackpot amount, 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 Jackpot which 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 subtract 40% 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 smaller prizes we assume that you don’t 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 partial results. $0.4682 + 0.0234 + 0.0083 + 0.1220 ~= $0.6218. Thus, even for a huge Jackpot with a quoted $400 million payout, 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. If you drive one mile to the store to buy your lottery ticket and then return home, you have driven two miles. Thus the probability that you will join this statistical group is 2 x 1.7 / 100,000,000 = 0.000000034. This can also be stated as “One 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 must lose. If we made a list of all the participants in a lottery, it might include:
1) Federal Government (Lottery winnings are taxable)
2) State Governments (Again lottery winnings are taxable)
3) State Governments (Direct share of lottery ticket sales)
4) Merchants that sell tickets (Paid by the lottery organizers)
5) 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:
1) Federal Government
2) State Government (Taxes)
3) State Government (Direct share)
4) Merchants that sell tickets
5) 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
Google’s/Yahoo’s search engines
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