Durango Bill's
Applied Mathematics
Powerball Odds
How to Calculate the Odds and Probabilities for the
Powerball Lottery
(Updated for the 59/35 game – New game as of 1/15/2012)
Applied Mathematics
Powerball Odds
How to Calculate the Odds and Probabilities for the
Powerball Lottery
(Updated for the 59/35 game – New game as of 1/15/2012)
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 35 numbers ranging from 1 to 35. 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” option. The multipliers in the 59/35 Power Play game increase the payout amounts for the non-jackpot prizes as shown in the “Power Play Option” section. (Scroll down the page.)
In the game version that began as of Jan. 15, 2012, it costs $2 to buy a ticket instead of the previous $1. The Power Play option costs another $1; and as noted above, the payout amounts have been changed.
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(35,1) = 35 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 35 = 175,223,510. 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: 1/175,223,510 = 0.000000005706996738+. If you express this as “One chance in ???”, you just divide “1” by the 0.000000005706996738+, which yields “One chance in 175,223,510”.
Match all 5 white balls but not the Powerball (Payout = $1,000,000)
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 34 losing Powerball numbers is: COMBIN(34,1) = 34. (Pick any of the 34 losers.) Thus there are COMBIN(5,5) x COMBIN(34,1) = 34 possible combinations. The probability for winning $1,000,000 is thus 34/175,223,510 = 0.000000194037889+ or “One chance in 5,153,632.65”.
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: 270/175,223,510 = 0.00000154088912- or “One chance in 648,975.96”.
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(34,1) = 34. The product of these is the number of ways you can win this configuration: COMBIN(5,4) x COMBIN(54,1) x COMBIN(34,1) = 9,180. The probability of success is thus: 9,180/175,223,510 = 0.00005239023+ or “One chance in 19,087.53”.
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: 14,310/175,223,510 = 0.000081667123+ or “One chance in 12,244.83”.
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(34,1) = 34. The product of these is the number of ways you can win this configuration: COMBIN(5,3) x COMBIN(54,2) x COMBIN(34,1) = 486,540. The probability of success is thus: 486,540/175,223,510 = 0.00277668+ or “One chance in 360.14”.
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(54,3) x COMBIN(1,1) = 248,040. The probability of success is thus: 248,040/175,223,510 = 0.00141556347+ or “One chance in 706.43”.
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(54,4) x COMBIN(1,1) = 1,581,255. The probability of success is thus: 1,581,255/175,223,510 = 0.009024217+ or “One chance in 110.81”.
Match 0 out of 5 white balls and match the Powerball (Payout = $4)
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: 3,162,510/175,223,510 = 0.018048434+ or “One chance in 55.41”.
Probability of winning something
If we add all the ways you can win something we get:
1 + 34 + 270 + 9,180 + 14,310 + 486,540 + 248,040 + 1,581,255 + 3,162,510 = 5,502,140. If we divide this number by 175,223,510, we get .031400695+ as a probability of winning something. 1 divided by 0.031400695+ yields “One chance in 31.85” 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 31.85. Thus, if the lottery officials proclaim that a given lottery drawing had 2 million “winners”, then there were about 2,000,000 x 31.85 ~= 63,692,858 tickets purchased that did not win the Jackpot. Alternately, there were about 63,692,858 - 2,000,000 ~= 61,692,858 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 0.0920 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 3/2. (Use the cash payout amount). For example, if the preceding game had a cash payout amount of $100,000,000 and the current game has a cash payout amount of $120,000,000, then there are about (3/2) x (120,000,000 – 100,000,000) = 30,000,000 tickets in play for the current game. (Each ticket sold for $2.) 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 7 8
-------------------------------------------------------------------------------------
20,000,000 0.8921 0.1018 0.0058 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000
40,000,000 0.7959 0.1817 0.0207 0.0016 0.0001 0.0000 0.0000 0.0000 0.0000
60,000,000 0.7101 0.2431 0.0416 0.0048 0.0004 0.0000 0.0000 0.0000 0.0000
80,000,000 0.6335 0.2892 0.0660 0.0100 0.0011 0.0001 0.0000 0.0000 0.0000
100,000,000 0.5651 0.3225 0.0920 0.0175 0.0025 0.0003 0.0000 0.0000 0.0000
120,000,000 0.5042 0.3453 0.1182 0.0270 0.0046 0.0006 0.0001 0.0000 0.0000
140,000,000 0.4498 0.3594 0.1436 0.0382 0.0076 0.0012 0.0002 0.0000 0.0000
160,000,000 0.4013 0.3664 0.1673 0.0509 0.0116 0.0021 0.0003 0.0000 0.0000
180,000,000 0.3580 0.3677 0.1889 0.0647 0.0166 0.0034 0.0006 0.0001 0.0000
200,000,000 0.3194 0.3645 0.2080 0.0792 0.0226 0.0052 0.0010 0.0002 0.0000
220,000,000 0.2849 0.3577 0.2246 0.0940 0.0295 0.0074 0.0016 0.0003 0.0000
240,000,000 0.2542 0.3482 0.2384 0.1089 0.0373 0.0102 0.0023 0.0005 0.0001
260,000,000 0.2268 0.3365 0.2496 0.1235 0.0458 0.0136 0.0034 0.0007 0.0001
280,000,000 0.2023 0.3233 0.2583 0.1376 0.0550 0.0176 0.0047 0.0011 0.0002
300,000,000 0.1805 0.3090 0.2645 0.1510 0.0646 0.0221 0.0063 0.0015 0.0003
320,000,000 0.1610 0.2941 0.2685 0.1635 0.0746 0.0273 0.0083 0.0022 0.0005
340,000,000 0.1436 0.2787 0.2704 0.1749 0.0848 0.0329 0.0106 0.0030 0.0007
360,000,000 0.1282 0.2633 0.2705 0.1852 0.0951 0.0391 0.0134 0.0039 0.0010
380,000,000 0.1143 0.2479 0.2689 0.1944 0.1054 0.0457 0.0165 0.0051 0.0014
400,000,000 0.1020 0.2328 0.2658 0.2022 0.1154 0.0527 0.0200 0.0065 0.0019
420,000,000 0.0910 0.2181 0.2614 0.2089 0.1252 0.0600 0.0240 0.0082 0.0025
440,000,000 0.0812 0.2039 0.2559 0.2142 0.1345 0.0675 0.0283 0.0101 0.0032
460,000,000 0.0724 0.1901 0.2496 0.2184 0.1433 0.0753 0.0329 0.0123 0.0041
480,000,000 0.0646 0.1770 0.2424 0.2214 0.1516 0.0831 0.0379 0.0148 0.0051
500,000,000 0.0576 0.1645 0.2347 0.2232 0.1592 0.0909 0.0432 0.0176 0.0063
520,000,000 0.0514 0.1526 0.2264 0.2240 0.1662 0.0986 0.0488 0.0207 0.0077
540,000,000 0.0459 0.1414 0.2179 0.2238 0.1724 0.1063 0.0546 0.0240 0.0093
560,000,000 0.0409 0.1308 0.2090 0.2227 0.1779 0.1137 0.0606 0.0277 0.0110
580,000,000 0.0365 0.1209 0.2000 0.2207 0.1826 0.1209 0.0667 0.0315 0.0131
600,000,000 0.0326 0.1115 0.1910 0.2180 0.1866 0.1278 0.0729 0.0357 0.0153
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 / 175,223,510 = 0.00000000571)
Pnotwin = (1.0 - Pwin) = 0.99999999429
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 .4498 0 0
1 .3594 120,000,000 43,124,542
2 .1436 60,000,000 8,613,907
3 .0382 40,000,000 1,529,408
4 .0076 30,000,000 229,119
5 .0012 24,000,000 29,290
6 .0002 20,000,000 3,250
Total 53,529,515
This total then has to be divided by 1 - .449787715 = .550212285 to give a weighted Jackpot amount of 53,529,515 / .550212285 ~= $97,288,840 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 0.7373 ~= $221,190,000 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 300,000,000 0.6266
20,000,000 0.9717 320,000,000 0.6063
40,000,000 0.9437 340,000,000 0.5866
60,000,000 0.9162 360,000,000 0.5676
80,000,000 0.8891 380,000,000 0.5491
100,000,000 0.8625 400,000,000 0.5313
120,000,000 0.8363 420,000,000 0.5141
140,000,000 0.8107 440,000,000 0.4974
160,000,000 0.7857 460,000,000 0.4814
180,000,000 0.7612 480,000,000 0.4659
200,000,000 0.7373 500,000,000 0.4510
220,000,000 0.7139 520,000,000 0.4366
240,000,000 0.6912 540,000,000 0.4228
260,000,000 0.6690 560,000,000 0.4095
280,000,000 0.6475 580,000,000 0.3967
300,000,000 0.6266 600,000,000 0.3844
The Powerball game includes an optional “Power Play”. If you spend an extra $1 for the “Power Play”, then the low order prizes are increased as shown in the following table.
Payout Payout Probability Expected
Match No Power Play With Power Play of result Value
5 for 5 not PB 1,000,000 2,000,000 1.94038E-07 0.1940
4 for 5 with PB 10,000 40,000 1.54089E-06 0.0462
4 for 5 not PB 100 200 5.23902E-05 0.0052
3 for 5 with PB 100 200 8.16671E-05 0.0082
3 for 5 not PB 7 14 0.002776682 0.0194
2 for 5 with PB 7 14 0.001415563 0.0099
2 for 5 not PB 4 12 0.009024217 0.0722
1 for 5 with PB 4 12 0.018048434 0.1444
Total 0.4996
Each row shows the combination involved, the payout amount to not include the Power Play, the payout amount with Power Play included, the probability of the particular output, and the expected value for this contribution. This “Expected Value” is the increase in payout amount times the probability. The total line shows that for each $1 that you spend for a Power Play option, you can expect to get back only $0.4996 – and this doesn’t include the taxes you would have to pay if you won $1,000,000.
It is interesting to calculate what the long term expected return is for each $2.00 lottery ticket that you buy.
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 $100,000,000 cash payout (decline the annuity) is assumed for the Jackpot. (Would be your portion of a shared Jackpot.)
Combination Payout Probability Contribution
-------------------------------------------------------
5 White + PB $100,000,000 5.70700E-09 $0.5707
5 White No PB 1,000,000 1.94038E-07 0.1940
4 White + PB 10,000 1.54089E-06 0.0154
4 White No PB 100 5.23902E-05 0.0052
3 White + PB 100 8.16671E-05 0.0082
3 White No PB 7 0.002776682 0.0194
2 White + PB 7 0.001415563 0.0099
1 White + PB 4 0.008098656 0.0361
PB 4 0.018048434 0.0722
Total 0.031400695 0.9312
Total for last 6 rows 0.1510
(Used for after tax calculation)
Thus, for each $2.00 that you spend for Powerball tickets, you can expect to get back about $0.9312. Of course you get to pay taxes on any large payout, so your net return is even less.
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 $600 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 $300,000,000. 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. We multiply the “$300,000,000 by the 0.7373 value from the 200,000,000 value in the above “Shared Jackpot” table to get $221,178,233 as the shared, before tax effective value of the Jackpot. However, this has to be reduced by 40% for taxes. This leaves an expected after tax value of $132,706,940. Finally, we multiply by the probability of winning (1 / 175,223,510) to get an expected after tax contribution from the Jackpot of $0.7574.
Next we include the after tax expected value from the two >= $10,000 prizes. This equals 0.1940+ 0.0154 = 0..2094 less 40% for taxes to give us an additional $0.1257-.
Finally, we add in the expected value for the “Total for last 6 rows” This adds another 0.1510 for our expected return. The sum of these three numbers is the expected after tax return for this particular combination. $0.7574 + $0.1257 + $ 0.1510 = $1.0341 expected after tax return for each $2 that you spend per ticket.
The average return per $ 2.00 ticket includes the extremely low probability that you might win a large prize – for example $10,000 or more. As a practical matter, it is unlikely that you will ever buy enough tickets (fork out enough money) to ever have much of a chance for any of the large prizes. Thus it is probable that all you will ever get back from your ticket purchases are piddling small amounts.
The percentages for these small amounts can be calculated. The table below shows the percentage chances for various “piddling returns”.
If you spend $2,000 to buy 1,000 tickets (1 ticket for each of 1,000 Powerball games), there is a:
50.09 % chance that you will get back $141 or less
60.36 % chance that you will get back $149 or less
69.94 % chance that you will get back $158 or less
79.92 % chance that you will get back $172 or less
89.99 % chance that you will get back $219 or less
95.08 % chance that you will get back $247 or less
98.02 % chance that you will get back $272 or less
99.00 % chance that you will get back $303 or less
99.49 % chance that you will get back $344 or less
99.80 % chance that you will get back $423 or less
Even if you buy 1,000 tickets, your chance of winning a $10,000 or larger prize is less than 0.2 %.
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 about 6 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 35 numbers ranging from 1 to 35. 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” option. The multipliers in the 59/35 Power Play game increase the payout amounts for the non-jackpot prizes as shown in the “Power Play Option” section. (Scroll down the page.)
In the game version that began as of Jan. 15, 2012, it costs $2 to buy a ticket instead of the previous $1. The Power Play option costs another $1; and as noted above, the payout amounts have been changed.
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(35,1) = 35 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 35 = 175,223,510. 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: 1/175,223,510 = 0.000000005706996738+. If you express this as “One chance in ???”, you just divide “1” by the 0.000000005706996738+, which yields “One chance in 175,223,510”.
Match all 5 white balls but not the Powerball (Payout = $1,000,000)
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 34 losing Powerball numbers is: COMBIN(34,1) = 34. (Pick any of the 34 losers.) Thus there are COMBIN(5,5) x COMBIN(34,1) = 34 possible combinations. The probability for winning $1,000,000 is thus 34/175,223,510 = 0.000000194037889+ or “One chance in 5,153,632.65”.
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: 270/175,223,510 = 0.00000154088912- or “One chance in 648,975.96”.
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(34,1) = 34. The product of these is the number of ways you can win this configuration: COMBIN(5,4) x COMBIN(54,1) x COMBIN(34,1) = 9,180. The probability of success is thus: 9,180/175,223,510 = 0.00005239023+ or “One chance in 19,087.53”.
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: 14,310/175,223,510 = 0.000081667123+ or “One chance in 12,244.83”.
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(34,1) = 34. The product of these is the number of ways you can win this configuration: COMBIN(5,3) x COMBIN(54,2) x COMBIN(34,1) = 486,540. The probability of success is thus: 486,540/175,223,510 = 0.00277668+ or “One chance in 360.14”.
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(54,3) x COMBIN(1,1) = 248,040. The probability of success is thus: 248,040/175,223,510 = 0.00141556347+ or “One chance in 706.43”.
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(54,4) x COMBIN(1,1) = 1,581,255. The probability of success is thus: 1,581,255/175,223,510 = 0.009024217+ or “One chance in 110.81”.
Match 0 out of 5 white balls and match the Powerball (Payout = $4)
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: 3,162,510/175,223,510 = 0.018048434+ or “One chance in 55.41”.
Probability of winning something
If we add all the ways you can win something we get:
1 + 34 + 270 + 9,180 + 14,310 + 486,540 + 248,040 + 1,581,255 + 3,162,510 = 5,502,140. If we divide this number by 175,223,510, we get .031400695+ as a probability of winning something. 1 divided by 0.031400695+ yields “One chance in 31.85” 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 31.85. Thus, if the lottery officials proclaim that a given lottery drawing had 2 million “winners”, then there were about 2,000,000 x 31.85 ~= 63,692,858 tickets purchased that did not win the Jackpot. Alternately, there were about 63,692,858 - 2,000,000 ~= 61,692,858 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 0.0920 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 3/2. (Use the cash payout amount). For example, if the preceding game had a cash payout amount of $100,000,000 and the current game has a cash payout amount of $120,000,000, then there are about (3/2) x (120,000,000 – 100,000,000) = 30,000,000 tickets in play for the current game. (Each ticket sold for $2.) 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 7 8
-------------------------------------------------------------------------------------
20,000,000 0.8921 0.1018 0.0058 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000
40,000,000 0.7959 0.1817 0.0207 0.0016 0.0001 0.0000 0.0000 0.0000 0.0000
60,000,000 0.7101 0.2431 0.0416 0.0048 0.0004 0.0000 0.0000 0.0000 0.0000
80,000,000 0.6335 0.2892 0.0660 0.0100 0.0011 0.0001 0.0000 0.0000 0.0000
100,000,000 0.5651 0.3225 0.0920 0.0175 0.0025 0.0003 0.0000 0.0000 0.0000
120,000,000 0.5042 0.3453 0.1182 0.0270 0.0046 0.0006 0.0001 0.0000 0.0000
140,000,000 0.4498 0.3594 0.1436 0.0382 0.0076 0.0012 0.0002 0.0000 0.0000
160,000,000 0.4013 0.3664 0.1673 0.0509 0.0116 0.0021 0.0003 0.0000 0.0000
180,000,000 0.3580 0.3677 0.1889 0.0647 0.0166 0.0034 0.0006 0.0001 0.0000
200,000,000 0.3194 0.3645 0.2080 0.0792 0.0226 0.0052 0.0010 0.0002 0.0000
220,000,000 0.2849 0.3577 0.2246 0.0940 0.0295 0.0074 0.0016 0.0003 0.0000
240,000,000 0.2542 0.3482 0.2384 0.1089 0.0373 0.0102 0.0023 0.0005 0.0001
260,000,000 0.2268 0.3365 0.2496 0.1235 0.0458 0.0136 0.0034 0.0007 0.0001
280,000,000 0.2023 0.3233 0.2583 0.1376 0.0550 0.0176 0.0047 0.0011 0.0002
300,000,000 0.1805 0.3090 0.2645 0.1510 0.0646 0.0221 0.0063 0.0015 0.0003
320,000,000 0.1610 0.2941 0.2685 0.1635 0.0746 0.0273 0.0083 0.0022 0.0005
340,000,000 0.1436 0.2787 0.2704 0.1749 0.0848 0.0329 0.0106 0.0030 0.0007
360,000,000 0.1282 0.2633 0.2705 0.1852 0.0951 0.0391 0.0134 0.0039 0.0010
380,000,000 0.1143 0.2479 0.2689 0.1944 0.1054 0.0457 0.0165 0.0051 0.0014
400,000,000 0.1020 0.2328 0.2658 0.2022 0.1154 0.0527 0.0200 0.0065 0.0019
420,000,000 0.0910 0.2181 0.2614 0.2089 0.1252 0.0600 0.0240 0.0082 0.0025
440,000,000 0.0812 0.2039 0.2559 0.2142 0.1345 0.0675 0.0283 0.0101 0.0032
460,000,000 0.0724 0.1901 0.2496 0.2184 0.1433 0.0753 0.0329 0.0123 0.0041
480,000,000 0.0646 0.1770 0.2424 0.2214 0.1516 0.0831 0.0379 0.0148 0.0051
500,000,000 0.0576 0.1645 0.2347 0.2232 0.1592 0.0909 0.0432 0.0176 0.0063
520,000,000 0.0514 0.1526 0.2264 0.2240 0.1662 0.0986 0.0488 0.0207 0.0077
540,000,000 0.0459 0.1414 0.2179 0.2238 0.1724 0.1063 0.0546 0.0240 0.0093
560,000,000 0.0409 0.1308 0.2090 0.2227 0.1779 0.1137 0.0606 0.0277 0.0110
580,000,000 0.0365 0.1209 0.2000 0.2207 0.1826 0.1209 0.0667 0.0315 0.0131
600,000,000 0.0326 0.1115 0.1910 0.2180 0.1866 0.1278 0.0729 0.0357 0.0153
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 / 175,223,510 = 0.00000000571)
Pnotwin = (1.0 - Pwin) = 0.99999999429
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 .4498 0 0
1 .3594 120,000,000 43,124,542
2 .1436 60,000,000 8,613,907
3 .0382 40,000,000 1,529,408
4 .0076 30,000,000 229,119
5 .0012 24,000,000 29,290
6 .0002 20,000,000 3,250
Total 53,529,515
This total then has to be divided by 1 - .449787715 = .550212285 to give a weighted Jackpot amount of 53,529,515 / .550212285 ~= $97,288,840 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 0.7373 ~= $221,190,000 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 300,000,000 0.6266
20,000,000 0.9717 320,000,000 0.6063
40,000,000 0.9437 340,000,000 0.5866
60,000,000 0.9162 360,000,000 0.5676
80,000,000 0.8891 380,000,000 0.5491
100,000,000 0.8625 400,000,000 0.5313
120,000,000 0.8363 420,000,000 0.5141
140,000,000 0.8107 440,000,000 0.4974
160,000,000 0.7857 460,000,000 0.4814
180,000,000 0.7612 480,000,000 0.4659
200,000,000 0.7373 500,000,000 0.4510
220,000,000 0.7139 520,000,000 0.4366
240,000,000 0.6912 540,000,000 0.4228
260,000,000 0.6690 560,000,000 0.4095
280,000,000 0.6475 580,000,000 0.3967
300,000,000 0.6266 600,000,000 0.3844
Power Play Option
The Powerball game includes an optional “Power Play”. If you spend an extra $1 for the “Power Play”, then the low order prizes are increased as shown in the following table.
Payout Payout Probability Expected
Match No Power Play With Power Play of result Value
5 for 5 not PB 1,000,000 2,000,000 1.94038E-07 0.1940
4 for 5 with PB 10,000 40,000 1.54089E-06 0.0462
4 for 5 not PB 100 200 5.23902E-05 0.0052
3 for 5 with PB 100 200 8.16671E-05 0.0082
3 for 5 not PB 7 14 0.002776682 0.0194
2 for 5 with PB 7 14 0.001415563 0.0099
2 for 5 not PB 4 12 0.009024217 0.0722
1 for 5 with PB 4 12 0.018048434 0.1444
Total 0.4996
Each row shows the combination involved, the payout amount to not include the Power Play, the payout amount with Power Play included, the probability of the particular output, and the expected value for this contribution. This “Expected Value” is the increase in payout amount times the probability. The total line shows that for each $1 that you spend for a Power Play option, you can expect to get back only $0.4996 – and this doesn’t include the taxes you would have to pay if you won $1,000,000.
Return on Investment
It is interesting to calculate what the long term expected return is for each $2.00 lottery ticket that you buy.
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 $100,000,000 cash payout (decline the annuity) is assumed for the Jackpot. (Would be your portion of a shared Jackpot.)
Combination Payout Probability Contribution
-------------------------------------------------------
5 White + PB $100,000,000 5.70700E-09 $0.5707
5 White No PB 1,000,000 1.94038E-07 0.1940
4 White + PB 10,000 1.54089E-06 0.0154
4 White No PB 100 5.23902E-05 0.0052
3 White + PB 100 8.16671E-05 0.0082
3 White No PB 7 0.002776682 0.0194
2 White + PB 7 0.001415563 0.0099
1 White + PB 4 0.008098656 0.0361
PB 4 0.018048434 0.0722
Total 0.031400695 0.9312
Total for last 6 rows 0.1510
(Used for after tax calculation)
Thus, for each $2.00 that you spend for Powerball tickets, you can expect to get back about $0.9312. Of course you get to pay taxes on any large payout, so your net return is even less.
Expected after tax return on your $2.00 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 $600 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 $300,000,000. 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. We multiply the “$300,000,000 by the 0.7373 value from the 200,000,000 value in the above “Shared Jackpot” table to get $221,178,233 as the shared, before tax effective value of the Jackpot. However, this has to be reduced by 40% for taxes. This leaves an expected after tax value of $132,706,940. Finally, we multiply by the probability of winning (1 / 175,223,510) to get an expected after tax contribution from the Jackpot of $0.7574.
Next we include the after tax expected value from the two >= $10,000 prizes. This equals 0.1940+ 0.0154 = 0..2094 less 40% for taxes to give us an additional $0.1257-.
Finally, we add in the expected value for the “Total for last 6 rows” This adds another 0.1510 for our expected return. The sum of these three numbers is the expected after tax return for this particular combination. $0.7574 + $0.1257 + $ 0.1510 = $1.0341 expected after tax return for each $2 that you spend per ticket.
Percentile Expected Returns on Ticket Purchases
The average return per $ 2.00 ticket includes the extremely low probability that you might win a large prize – for example $10,000 or more. As a practical matter, it is unlikely that you will ever buy enough tickets (fork out enough money) to ever have much of a chance for any of the large prizes. Thus it is probable that all you will ever get back from your ticket purchases are piddling small amounts.
The percentages for these small amounts can be calculated. The table below shows the percentage chances for various “piddling returns”.
If you spend $2,000 to buy 1,000 tickets (1 ticket for each of 1,000 Powerball games), there is a:
50.09 % chance that you will get back $141 or less
60.36 % chance that you will get back $149 or less
69.94 % chance that you will get back $158 or less
79.92 % chance that you will get back $172 or less
89.99 % chance that you will get back $219 or less
95.08 % chance that you will get back $247 or less
98.02 % chance that you will get back $272 or less
99.00 % chance that you will get back $303 or less
99.49 % chance that you will get back $344 or less
99.80 % chance that you will get back $423 or less
Even if you buy 1,000 tickets, your chance of winning a $10,000 or larger prize is less than 0.2 %.
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 about 6 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
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