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. How to calculate these
Powerball odds. Jackpot split probabilities including return on investment calculations.

Ticket Matches Payout Odds Probability

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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.

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 sales slumped. Thus Powerball officials changed 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 https://en.wikipedia.org/wiki/There's_a_sucker_born_every_minute )

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.

(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.)

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:

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

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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 )

For this example we will assume the cash value of the Jackpot is $600,000,000 and there are 200,000,000 tickets in play for the current game. Probability values are from the “200,000,000” row above.

The first calculation is: “What is the probability that the jackpot will be won?” This is simply (1.00 – the probability that no one will win) = 1.00 – 0.5044 = 0.4956. Thus the expected jackpot payout by the lottery is $600,000,000 times 0.4956 = $297,382,360.

If there are 200,000,000 tickets in play, then we divide the $297,382,360 by 200,000,000 to get an average jackpot payout per ticket of $1.49. The other smaller prizes add $0.3199 to this amount to give an "expected before tax, cash value of $1.81.

These calculations can be used to form a table that shows the expected return per ticket ( = expected value per ticket). For example if the cash value of the jackpot is $600,000,000 and there are 200,000,000 tickets in play, then the ticket’s expected value is $1.81.

The following table shows the "Expected Before Taxes Value" (includes $0.3199 for the smaller prizes) of a $2.00 ticket.

Nbr. Tickets

In Play < - - - - Cash (not annuity) Jackpot Size in Millions - - - - >

In Millions 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300

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100 0.61 0.90 1.19 1.48 1.77 2.06 2.35 2.64 2.93 3.22 3.51 3.80 4.09

200 0.57 0.82 1.06 1.31 1.56 1.81 2.05 2.30 2.55 2.80 3.05 3.29 3.54

300 0.53 0.75 0.96 1.18 1.39 1.60 1.82 2.03 2.25 2.46 2.67 2.89 3.10

400 0.51 0.69 0.88 1.07 1.25 1.44 1.62 1.81 2.00 2.18 2.37 2.56 2.74

500 0.48 0.65 0.81 0.98 1.14 1.30 1.47 1.63 1.79 1.96 2.12 2.29 2.45

600 0.47 0.61 0.76 0.90 1.05 1.19 1.34 1.48 1.63 1.77 1.92 2.06 2.21

700 0.45 0.58 0.71 0.84 0.97 1.10 1.23 1.36 1.49 1.62 1.75 1.88 2.01

800 0.44 0.55 0.67 0.79 0.90 1.02 1.14 1.26 1.37 1.49 1.61 1.72 1.84

900 0.43 0.53 0.64 0.74 0.85 0.96 1.06 1.17 1.27 1.38 1.49 1.59 1.70

1000 0.42 0.51 0.61 0.71 0.80 0.90 1.00 1.09 1.19 1.29 1.38 1.48 1.58

We can also see what happens to the expected value of a ticket if a buying frenzy should develop at this point. Let’s assume that 300 million more tickets are sold. At $2,00 per ticket, the lottery takes in $600 million. 1/2 of this goes into the total prize pot. (1/3 for the jackpot and 1/6 for the smaller prizes.) The jackpot is now worth $600 million plus $200 million = $800 million.

Thus the game is transformed into 500 million tickets in play for a cash jackpot that is now worth $800 million. If we follow the 500-million row to the right until we reach the $800 million column, we find an expected cash value of $1.63. The buying frenzy has reduced the expected value of a ticket from $1.81 to $1.63.

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 Probability Probability

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.

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

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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.

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, we check the expected value of a ticket in the table that we calculated earlier. Follow the 600-million row until you come to the $1,000 million column. The expected cash value of the ticket is $1.77.This included $0.3199 for the smaller prizes so $0.3199 has to be subtracted back out. This leaves $1.45 for the Jackpot component. However, this has to be reduced by 40% for taxes. This leaves an expected after tax value of the jackpot of $0.8717.

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. $0.8717 + $0.0842 + $ 0.1796 = $1.1355 expected after tax return for each $2 that you spend per ticket.

The expected value of a ticket will vary depending on how many tickets are in play (shared jackpot calculations) as well as the payout rules for a game. If the number of tickets in play were proportional to the size of the jackpot, then the expected value of a ticket would asymptotically approach (gradually approach but never quite reach) some fixed value. (Proportional increase = If the size of the jackpot doubled, then the number of tickets in play would double.)

In practice, buying frenzies develop when a large jackpot exists – particularly if the quoted jackpot is larger than any previous jackpot. Thus the number of tickets in play increases faster than a simple proportional increase. When this happens, the expected value of a ticket will actually decrease when huge jackpots exist. We saw this happen with the very large “Billion dollar” jackpot in mid Jan. 2016. The more the jackpot increases, the greater the buying frenzy. The result is that the expected value of a ticket is actually decreasing even though the quoted size of the jackpot is increasing.

Thus the general shape of a graph plotting of the “expected value” of a ticket will resemble the shape of the red line in the graph shown earlier.

This principle of decreasing “expected value” can be illustrated by 2 simple examples. We will assume that the only prize is the jackpot. Also 1/3 of any new money that is spent on tickets is used to increase the jackpot. (This is the way that Powerball is actually run.) In both cases we will assume that a $1 billion cash value jackpot exists prior to ticket purchases.

Example 1)

One ticket is purchased. Net proceeds to the lottery are 1 x $2.00 = $2.00. 1/3 of this is 2 divided by 3 = $0.67. This is added to the cash value of the lottery which brings the jackpot up to $1,000,000,000.67. If only 1 ticket is in play, then no adjustment has to be made for splitting the jackpot. The before tax expected value of the ticket is the value of the jackpot divided by the number of combinations. This becomes $1,000,000,000.67 divided by 292,201,338 = $3.42. This has to be reduced by 40% to get the after tax expected value. $3.42 less 40% = $2.05. The expected after tax value of the ticket is $2.05 - which would be marginaly profitable.

Example 2)

One billion (1,000,000,000) tickets are purchased. Net proceeds to the lottery are 1,000,000,000 x $2.00 = $2,000,000,000.00. 1/3 of this is 2,000,000,000 divided by 3 = $666,666,666.67. This is added to the cash value of the lottery which brings the jackpot up to $1,666,666,666.67. There is a 0.9674 probability that at least one winner will exist if 1,000,000,000 tickets in play. If there are 1,000,000,000 tickets in play, then the expected jackpot per ticket is $1.6123. The $1.6123 payout has to be reduced by another 40% to give an after tax return. Thus brings the after tax expected share of the jackpot down to $0.9674. You spent $2.00 for an expected after tax return of $0.97.

The average return per $ 2.00 ticket includes the extremely low probability that you might win a large prize – for example $50,000 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:

49.67 % chance that you will get back $172 or less

59.94 % chance that you will get back $180 or less

69.98 % chance that you will get back $189 or less

79.76 % chance that you will get back $201 or less

90.08 % chance that you will get back $231 or less

94.97 % chance that you will get back $268 or less

97.98 % chance that you will get back $295 or less

99.00 % chance that you will get back $314 or less

99.50 % chance that you will get back $346 or less

99.88 % chance that you will get back $504 or less

Even if you buy 1,000 tickets, your chance of winning a $50,000 or larger prize is less than 0.12 %.

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 10 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 79 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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Updated for the 69/26 Game

Concise Table of Powerball Odds (Mathematical derivation below)

Ticket Matches Payout Odds Probability

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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 sales slumped. Thus Powerball officials changed 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 https://en.wikipedia.org/wiki/There's_a_sucker_born_every_minute )

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.)

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:

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

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

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 Ticket Value

Considering the Number of Tickets that are in Play

Considering the Number of Tickets that are in Play

For this example we will assume the cash value of the Jackpot is $600,000,000 and there are 200,000,000 tickets in play for the current game. Probability values are from the “200,000,000” row above.

The first calculation is: “What is the probability that the jackpot will be won?” This is simply (1.00 – the probability that no one will win) = 1.00 – 0.5044 = 0.4956. Thus the expected jackpot payout by the lottery is $600,000,000 times 0.4956 = $297,382,360.

If there are 200,000,000 tickets in play, then we divide the $297,382,360 by 200,000,000 to get an average jackpot payout per ticket of $1.49. The other smaller prizes add $0.3199 to this amount to give an "expected before tax, cash value of $1.81.

These calculations can be used to form a table that shows the expected return per ticket ( = expected value per ticket). For example if the cash value of the jackpot is $600,000,000 and there are 200,000,000 tickets in play, then the ticket’s expected value is $1.81.

The following table shows the "Expected Before Taxes Value" (includes $0.3199 for the smaller prizes) of a $2.00 ticket.

Nbr. Tickets

In Play < - - - - Cash (not annuity) Jackpot Size in Millions - - - - >

In Millions 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300

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

100 0.61 0.90 1.19 1.48 1.77 2.06 2.35 2.64 2.93 3.22 3.51 3.80 4.09

200 0.57 0.82 1.06 1.31 1.56 1.81 2.05 2.30 2.55 2.80 3.05 3.29 3.54

300 0.53 0.75 0.96 1.18 1.39 1.60 1.82 2.03 2.25 2.46 2.67 2.89 3.10

400 0.51 0.69 0.88 1.07 1.25 1.44 1.62 1.81 2.00 2.18 2.37 2.56 2.74

500 0.48 0.65 0.81 0.98 1.14 1.30 1.47 1.63 1.79 1.96 2.12 2.29 2.45

600 0.47 0.61 0.76 0.90 1.05 1.19 1.34 1.48 1.63 1.77 1.92 2.06 2.21

700 0.45 0.58 0.71 0.84 0.97 1.10 1.23 1.36 1.49 1.62 1.75 1.88 2.01

800 0.44 0.55 0.67 0.79 0.90 1.02 1.14 1.26 1.37 1.49 1.61 1.72 1.84

900 0.43 0.53 0.64 0.74 0.85 0.96 1.06 1.17 1.27 1.38 1.49 1.59 1.70

1000 0.42 0.51 0.61 0.71 0.80 0.90 1.00 1.09 1.19 1.29 1.38 1.48 1.58

We can also see what happens to the expected value of a ticket if a buying frenzy should develop at this point. Let’s assume that 300 million more tickets are sold. At $2,00 per ticket, the lottery takes in $600 million. 1/2 of this goes into the total prize pot. (1/3 for the jackpot and 1/6 for the smaller prizes.) The jackpot is now worth $600 million plus $200 million = $800 million.

Thus the game is transformed into 500 million tickets in play for a cash jackpot that is now worth $800 million. If we follow the 500-million row to the right until we reach the $800 million column, we find an expected cash value of $1.63. The buying frenzy has reduced the expected value of a ticket from $1.81 to $1.63.

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 Probability Probability

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

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, we check the expected value of a ticket in the table that we calculated earlier. Follow the 600-million row until you come to the $1,000 million column. The expected cash value of the ticket is $1.77.This included $0.3199 for the smaller prizes so $0.3199 has to be subtracted back out. This leaves $1.45 for the Jackpot component. However, this has to be reduced by 40% for taxes. This leaves an expected after tax value of the jackpot of $0.8717.

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. $0.8717 + $0.0842 + $ 0.1796 = $1.1355 expected after tax return for each $2 that you spend per ticket.

General Rules for the Expected Value of a Ticket

The expected value of a ticket will vary depending on how many tickets are in play (shared jackpot calculations) as well as the payout rules for a game. If the number of tickets in play were proportional to the size of the jackpot, then the expected value of a ticket would asymptotically approach (gradually approach but never quite reach) some fixed value. (Proportional increase = If the size of the jackpot doubled, then the number of tickets in play would double.)

In practice, buying frenzies develop when a large jackpot exists – particularly if the quoted jackpot is larger than any previous jackpot. Thus the number of tickets in play increases faster than a simple proportional increase. When this happens, the expected value of a ticket will actually decrease when huge jackpots exist. We saw this happen with the very large “Billion dollar” jackpot in mid Jan. 2016. The more the jackpot increases, the greater the buying frenzy. The result is that the expected value of a ticket is actually decreasing even though the quoted size of the jackpot is increasing.

Thus the general shape of a graph plotting of the “expected value” of a ticket will resemble the shape of the red line in the graph shown earlier.

This principle of decreasing “expected value” can be illustrated by 2 simple examples. We will assume that the only prize is the jackpot. Also 1/3 of any new money that is spent on tickets is used to increase the jackpot. (This is the way that Powerball is actually run.) In both cases we will assume that a $1 billion cash value jackpot exists prior to ticket purchases.

Example 1)

One ticket is purchased. Net proceeds to the lottery are 1 x $2.00 = $2.00. 1/3 of this is 2 divided by 3 = $0.67. This is added to the cash value of the lottery which brings the jackpot up to $1,000,000,000.67. If only 1 ticket is in play, then no adjustment has to be made for splitting the jackpot. The before tax expected value of the ticket is the value of the jackpot divided by the number of combinations. This becomes $1,000,000,000.67 divided by 292,201,338 = $3.42. This has to be reduced by 40% to get the after tax expected value. $3.42 less 40% = $2.05. The expected after tax value of the ticket is $2.05 - which would be marginaly profitable.

Example 2)

One billion (1,000,000,000) tickets are purchased. Net proceeds to the lottery are 1,000,000,000 x $2.00 = $2,000,000,000.00. 1/3 of this is 2,000,000,000 divided by 3 = $666,666,666.67. This is added to the cash value of the lottery which brings the jackpot up to $1,666,666,666.67. There is a 0.9674 probability that at least one winner will exist if 1,000,000,000 tickets in play. If there are 1,000,000,000 tickets in play, then the expected jackpot per ticket is $1.6123. The $1.6123 payout has to be reduced by another 40% to give an after tax return. Thus brings the after tax expected share of the jackpot down to $0.9674. You spent $2.00 for an expected after tax return of $0.97.

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 $50,000 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:

49.67 % chance that you will get back $172 or less

59.94 % chance that you will get back $180 or less

69.98 % chance that you will get back $189 or less

79.76 % chance that you will get back $201 or less

90.08 % chance that you will get back $231 or less

94.97 % chance that you will get back $268 or less

97.98 % chance that you will get back $295 or less

99.00 % chance that you will get back $314 or less

99.50 % chance that you will get back $346 or less

99.88 % chance that you will get back $504 or less

Even if you buy 1,000 tickets, your chance of winning a $50,000 or larger prize is less than 0.12 %.

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 10 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 79 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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