Updated for the 69/26 Game

Durango Bill's
Applied Mathematics
Powerball Odds
How to Calculate the Odds and Probabilities for the
Powerball Lottery

Powerball odds and probabilities for the Powerball Jackpot – and how to calculate these Powerball odds. Additional calculations show tie probabilities and expected return on your “investment”.

Concise Table of Powerball Odds (Mathematical derivation below)

Ticket Matches Payout Odds Probability
------
5 White + PB Jackpot 1 in 292,201,338.00 0.000000003422
5 White No PB 1,000,000 1 in 11,688,053.52 0.00000008556
4 White + PB 50,000 1 in 913,129.18 0.000001095
4 White No PB 100 1 in 36,525.17 0.00002738
3 White + PB 100 1 in 14,494.11 0.00006899
3 White No PB 7 1 in 579.76 0.001725
2 White + PB 7 1 in 701.33 0.001426
1 White + PB 4 1 in 91.98 0.01087
0 White + PB 4 1 in 38.32 0.02609
Win something Variable 1 in 24.87 0.0402
Game Rules
The numbers picked for the prizes consist of 5 white balls picked at random from a drum that holds 69 balls numbered from 1 to 69. The Powerball number is a single ball that is picked from a second drum that has 26 numbers ranging from 1 to 26. 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 69/26 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.

Game Rules for the new Powerball game that went into effect for the Oct. 7, 2015 game.

As “game players” (“suckers”) woke up to the fact that they were throwing money away trying to win the old 59/35 game, Powerball ticket salesslumped. Thus Powerball officialschanged the game rules again to try to recruit more people to throw away their money.
The new game is designed to “engineer” bigger jackpots. The mechanism involved was to make it even more difficult to win. Thus funds that previously had been paid out to “millionaire” winners will now be retained until a possible “billionaire” figure is reached.
In the old version of the game, the chance of winning the jackpot was one chance in COMBIN(59,5) x COMBIN(35,1) = 175,223,510. The new version of the game has 69 balls in one bin and 26 in the other. Thus the chance of winning the new game is 1 chance in COMBIN(69,5) x COMBIN(26,1) = 292,201,338. In practical terms, it would appear likely that few people will buy tickets for most jackpots, but buying frenzies will develop for large jackpots. (With the resulting prize split several ways.)
Imagine lining up baseballs (A standard baseball is about 2.9 inches in diameter.) in a row for the 2,998.68 highway miles from Boston to Los Angeles (Mapquest). It would take about 65,515,988 baseballs. Then randomly designate one of these baseballs as a lucky “winner” baseball.
Imagine driving for days past this row of millions and millions of baseballs. Then stop and pick up a random baseball. The chance of a random ticket winning the new Powerball is less than one fourth the chance of picking the winning baseball.
The phrase “There's a sucker born every minute” comes to mind. (Falsely attributed to P. T. Barnum )
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 69 is: COMBIN(69,5) = 11,238,513. (See the math notation page or Help in Microsoft's Excel for more information on “COMBIN”).
For each of these 11,238,513 combinations there are COMBIN(26,1) = 26 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 11,238,513 x 26 = 292,201,338. 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: 1/292,201,338 = 0.000000003422297813+. If you express this as “One chance in ???”, you just divide “1” by the 0.000000003422297813+, which yields “One chance in 292,201,338”.
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 25 losing Powerball numbers is: COMBIN(25,1) = 25. (Pick any of the 25 losers.) Thus there are COMBIN(5,5) x COMBIN(25,1) = 25 possible combinations. The probability for winning $1,000,000 is thus 25/292,201,338 ~= 0.00000008556 or “One chance in 11,688,053.52”.
Match 4 out of 5 white balls and match the Powerball (Payout = $50,000)
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 64 losing white numbers is COMBIN(64,1) = 64. 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(64,1) x COMBIN(1,1) = 320. The probability of success is thus: 320/292,201,338 ~= 0.000001095 or “One chance in 913,129.18”.
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 64 losing numbers is COMBIN(64,1) = 64. The number of ways your Powerball number can miss matching the single Powerball number is: COMBIN(25,1) = 25. The product of these is the number of ways you can win this configuration:COMBIN(5,4) x COMBIN(64,1) x COMBIN(25,1) = 8,000. The probability of success is thus: 8,000/292,201,338 ~= 0.00002738 or “One chance in 36,525.17”.
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 64 losing white numbers is COMBIN(64,2) = 2,016. 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(64,2) x COMBIN(1,1) = 20,160. The probability of success is thus: 20,160/292,201,338 ~= 0.00006899 or “One chance in 14,494.11”.
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 64 losing numbers is COMBIN(64,2) = 2,016. The number of ways your Powerball number can miss matching the single Powerball number is: COMBIN(25,1) = 25. The product of these is the number of ways you can win this configuration:COMBIN(5,3) x COMBIN(64,2) x COMBIN(25,1) = 504,000. The probability of success is thus: 504,000/292,201,338 ~= 0.001725 or “One chance in 579.76”.
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 64 losing white numbers is COMBIN(64,3) = 41,664. 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(64,3) x COMBIN(1,1) = 416,640. The probability of success is thus: 416,640/292,201,338 ~= 0.001426 or “One chance in 701.33”.
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 64 losing white numbers is COMBIN(64,4) = 635,376. 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(64,4) x COMBIN(1,1) = 3,176,880. The probability of success is thus: 3,176,880/292,201,338 ~= 0.01087 or “One chance in 91.98”.
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 64 losing white numbers is COMBIN(64,5) = 7,624,512. 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(64,5) x COMBIN(1,1) = 7,624,512. The probability of success is thus: 7,624,512/292,201,338 ~= 0.02609 or “One chance in 38.32”.
Probability of winning something
If we add all the ways you can win something we get:
1 + 25 + 320 + 8,000 + 20,160 + 504,000 + 416,640 + 3,176,880 + 7,624,512 = 11,750,538. If we divide this number by 292,201,338, we get .04021+ as a probability of winning something. 1 divided by 0.04021- yields “One chance in 24.87” 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 24.87. Thus, if the lottery officials proclaim that a given lottery drawing had 3 million “winners”, then there were about 3,000,000 x 24.87 ~= 74,601,181 tickets purchased that did not win the Jackpot. Alternately, there were about 74,601,181 - 3,000,000 ~= 71,601,181 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 1-69 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.)

Chart shows probabilities of 0 1 2 or more winners

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 0.0416 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:

“N” Number “K”
of tickets Number of tickets holding the Jackpot combination
in play 0 1 2 3 4 5 6 7 8
------
100,000,000 0.7102 0.2430 0.0416 0.0047 0.0004 0.0000 0.0000 0.0000 0.0000
200,000,000 0.5044 0.3452 0.1181 0.0270 0.0046 0.0006 0.0001 0.0000 0.0000
300,000,000 0.3582 0.3678 0.1888 0.0646 0.0166 0.0034 0.0006 0.0001 0.0000
400,000,000 0.2544 0.3482 0.2383 0.1088 0.0372 0.0102 0.0023 0.0005 0.0001
500,000,000 0.1807 0.3091 0.2645 0.1509 0.0645 0.0221 0.0063 0.0015 0.0003
600,000,000 0.1283 0.2634 0.2705 0.1851 0.0950 0.0390 0.0134 0.0039 0.0010
700,000,000 0.0911 0.2183 0.2615 0.2088 0.1250 0.0599 0.0239 0.0082 0.0025
800,000,000 0.0647 0.1772 0.2425 0.2213 0.1515 0.0830 0.0379 0.0148 0.0051
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 / 292,201,338 = 0.00000000342)
Pnotwin = (1.0 - Pwin) = 0.99999999658
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 $240,000,000 and there are 200,000,000 tickets in play for the current game. Probability values are from the “200,000,000” row above.
Number of Jackpot paid Contribution
winners Probability to each winner (Col 2 x Col 3)
------
0 .5044 0 0
1 .3452 240,000,000 82,851,815
2 .1181 120,000,000 14,177,179
3 .0270 80,000,000 2,156,379
4 .0046 60,000,000 276,741
5 .0006 48,000,000 30,307
6 .0001 40,000,000 2,881
Other shares 261
Total 99,495,564
This total then has to be divided by 1 - .5043627 = .495637 to give a weighted Jackpot amount of 99,495,564 / .495637 ~= $200,742,702 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 0.8364 ~= $250,928,378 to adjust for the possibility that a winning ticket will have to split the Jackpot with some other winning ticket.
Number ofMult. Jackpot by Number ofMult. Jackpot by
Tickets this ratio for Tickets this ratio for
in play possible sharing in play possible sharing
0 1.0000 400,000,000 0.6913
50,000,000 0.9576 450,000,000 0.6584
100,000,000 0.9162 500,000,000 0.6268
150,000,000 0.8758 550,000,000 0.5966
200,000,000 0.8364 600,000,000 0.5678
250,000,000 0.7983 650,000,000 0.5403
300,000,000 0.7613 700,000,000 0.5143
350,000,000 0.7257 750,000,000 0.4895
400,000,000 0.6913 800,000,000 0.4661

Power Play Option (assume 10x 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.
The Power Play has a random multiplier as per the following table.
Multiplier times
Multiplier ProbabilityProbability
2X 24/43 1.1163
3X 13/43 0.9070
4X 3/43 0.2791
5X 2/43 0.2326
10X 1/43 0.2326
Sum 2.7674
Thus the expected average total payout if you pay for the Power Play option is 2.7674 times the original payouts. Since you would get the original payouts without paying for the Power Play option, the net value of the Power Play is the increase in payout amounts. This increase in payout amounts is: 2.7674 – 1.0 = 1.7674 times the original payout amounts. We can use this 1.7674 multiplier to calculate the expected return if you pay the extra $1.00 for the Power Play option.
Payout Increased Exp. Val
Without Payout With Probability Expected After
Match Power Play Power Play of result Value Taxes
5 for 5 not PB 1,000,000 = 1,000,000 8.55574E-08 0.0856 0.0513
4 for 5 with PB 50,000 x 1.7674 = 88,372.09 1.09514E-06 0.0968 0.0581
4 for 5 not PB 100 x 1.7674 = 176.74 2.73784E-05 0.0048 0.0048
3 for 5 with PB 100 x 1.7674 = 176.74 6.89935E-05 0.0122 0.0122
3 for 5 not PB 7 x 1.7674 = 12.37 0.001724838 0.0213 0.0213
2 for 5 with PB 7 x 1.7674 = 12.37 0.001425866 0.0176 0.0176
2 for 5 not PB 4 x 1.7674 = 7.07 0.010872229 0.0769 0.0769
1 for 5 with PB 4 x 1.7674 = 7.07 0.026093351 0.1845 0.1845
Total 0.4997 0.4268
Each row shows the combination involved, the payout amount without including the Power Play, the increased payout amount with Power Play included, the probability of the particular output, the expected value for this contribution, and the expected value after 40% is deducted for federal, state, and local taxes. The “Expected Value” is the increase in payout amount times the probability. The total line shows that for each $1.00 that you spend for a Power Play option, you can expect to get back only $0.4997. Taxes reduce this long term expected payout to less than $0.43 for each dollar you pay for the Power Play.
An analysis for Power Play without the 10X option shows the same approximate $0.50 per $1.00 spent return.

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 $200,000,000 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 $200,000,000 3.42230E-09 $0.6845
5 White No PB 1,000,000 8.55574E-08 0.0856
4 White + PB 50,000 1.09514E-06 0.0548
4 White No PB 100 2.73784E-05 0.0027
3 White + PB 100 6.89935E-05 0.0069
3 White No PB 7 0.001724838 0.0121
2 White + PB 7 0.001425866 0.0100
1 White + PB 4 0.010872229 0.0435
PB 4 0.026093351 0.1044
Total 0.040213840 1.0043
Total for last 6 rows 0.1796
(Used for after tax calculation)
Thus, for each $2.00 that you spend for Powerball tickets, you can expect to get back about $1.0043. 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 huge 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 huge Jackpot exists. The following analysis assumes the annuity value of the Jackpot is $2 Billion (that’s a “B”) and there are 600 million tickets in play. The cash value for any Jackpot is about one-half the annuity value which brings the real value down to $1,000,000,000. All prizes of $50,000 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 600 million tickets in play. We multiply the “$1,000,000,000 by the 0.5678 value from the 600,000,000 row in the above “Shared Jackpot” table to get $567,800,000 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 ~$340,680,000. Finally, we multiply by the probability of winning (1 / 292,201,338) to get an expected after tax contribution from the Jackpot of $1.1658.
Next we include the after tax expected value from the two >= $50,000 prizes. This equals 0.0856+ 0.0548 = 0.1403 less 40% for taxes to give us an additional $0.0842.
Finally, we add in the expected value for the “Total for last 6 rows” This adds another 0.1796 for our expected return. The sum of these three numbers is the expected after tax return for this particular combination. $1.1658 + $0.0842 + $ 0.1796 = $1.4296 expected after tax return for each $2 that you spend per ticket.