Durango Bill's

Applied Mathematics

Powerball Odds

How to Calculate the Odds and Probabilities for the

Powerball Lottery

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”.

Ticket Matches Payout Odds Probability

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

5 White + PB Jackpot 1 in 175,223,510 0.000000005707

5 White No PB 1,000,000 1 in 5,153,632.65 0.0000001940

4 White + PB 10,000 1 in 648,975.96 0.000001541

4 White No PB 100 1 in 19,087.53 0.00005239

3 White + PB 100 1 in 12,244.83 0.00008167

3 White No PB 7 1 in 360.14 0.002777

2 White + PB 7 1 in 706.43 0.001416

1 White + PB 4 1 in 110.81 0.009024

0 White + PB 4 1 in 55.41 0.01805

Win something Variable 1 in 31.85 0.03140

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.

The Power Play has returned to a random multiplier as per the following table.

Multiplier times

Multiplier Probability Probability

2X 0.50 1.0

3X 0.30 0.9

4X 0.10 0.4

5X 0.10 0.5

Sum 2.8

Thus the expected average total payout if you pay for the Power Play option is 2.8 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: 2.8 – 1.0 = 1.8 times the original payout amounts. We can use this 1.8 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 x 1.8 = 1,800,000 1.94038E-07 0.3493 0.2096

4 for 5 with PB 10,000 x 1.8 = 18,000 1.54089E-06 0.0277 0.0166

4 for 5 not PB 100 x 1.8 = 180 5.23902E-05 0.0094 0.0094

3 for 5 with PB 100 x 1.8 = 180 8.16671E-05 0.0147 0.0147

3 for 5 not PB 7 x 1.8 = 12.60 0.002776682 0.0350 0.0350

2 for 5 with PB 7 x 1.8 = 12.60 0.001415563 0.0178 0.0178

2 for 5 not PB 4 x 1.8 = 7.20 0.009024217 0.0144 0.0144

1 for 5 with PB 4 x 1.8 = 7.20 0.018048434 0.0289 0.0289

Total 0.4973 0.3465

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 $0.4973. Taxes reduce this long term expected payout to less than $0.35 for each dollar you pay for the Power Play.

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.

Alternately, if you “played” Russian Roulette 100 times per day every day for 48 years with Powerball Jackpot odds, you would have better than a 99% chance of surviving.

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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Concise Table of Powerball Odds (Mathematical derivation below)

Ticket Matches Payout Odds Probability

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

5 White + PB Jackpot 1 in 175,223,510 0.000000005707

5 White No PB 1,000,000 1 in 5,153,632.65 0.0000001940

4 White + PB 10,000 1 in 648,975.96 0.000001541

4 White No PB 100 1 in 19,087.53 0.00005239

3 White + PB 100 1 in 12,244.83 0.00008167

3 White No PB 7 1 in 360.14 0.002777

2 White + PB 7 1 in 706.43 0.001416

1 White + PB 4 1 in 110.81 0.009024

0 White + PB 4 1 in 55.41 0.01805

Win something Variable 1 in 31.85 0.03140

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.

The Power Play has returned to a random multiplier as per the following table.

Multiplier times

Multiplier Probability Probability

2X 0.50 1.0

3X 0.30 0.9

4X 0.10 0.4

5X 0.10 0.5

Sum 2.8

Thus the expected average total payout if you pay for the Power Play option is 2.8 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: 2.8 – 1.0 = 1.8 times the original payout amounts. We can use this 1.8 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 x 1.8 = 1,800,000 1.94038E-07 0.3493 0.2096

4 for 5 with PB 10,000 x 1.8 = 18,000 1.54089E-06 0.0277 0.0166

4 for 5 not PB 100 x 1.8 = 180 5.23902E-05 0.0094 0.0094

3 for 5 with PB 100 x 1.8 = 180 8.16671E-05 0.0147 0.0147

3 for 5 not PB 7 x 1.8 = 12.60 0.002776682 0.0350 0.0350

2 for 5 with PB 7 x 1.8 = 12.60 0.001415563 0.0178 0.0178

2 for 5 not PB 4 x 1.8 = 7.20 0.009024217 0.0144 0.0144

1 for 5 with PB 4 x 1.8 = 7.20 0.018048434 0.0289 0.0289

Total 0.4973 0.3465

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 $0.4973. Taxes reduce this long term expected payout to less than $0.35 for each dollar you pay for the Power Play.

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.

Alternately, if you “played” Russian Roulette 100 times per day every day for 48 years with Powerball Jackpot odds, you would have better than a 99% chance of surviving.

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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