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powerball expected value

We did the math to see if it’s worth buying a Powerball lottery ticket

That is a pretty huge chunk of money. However, as we saw before Saturday’s drawing when the jackpot was $535 million, taking a closer look at the underlying math of the lottery shows that it’s probably a bad idea to buy a ticket.

Consider the expected value

When trying to evaluate the outcome of a risky, probabilistic event like the lottery, one of the first things to look at is expected value.

The expected value of a randomly decided process is found by taking all the possible outcomes of the process, multiplying each outcome by its probability, and adding all those numbers up. This gives us a long-run average value for our random process.

Expected value is helpful for assessing gambling outcomes. If my expected value for playing the game, based on the cost of playing and the probabilities of winning different prizes, is positive, then, in the long run, the game will make me money. If expected value is negative, then this game is a net loser for me.

Lotteries are a great example of this kind of probabilistic process. In Powerball, for each $2 ticket you buy, you choose five numbers between 1 and 69 (represented by white balls in the drawing) and one number between 1 and 26 (the red “powerball”). Prizes are based on how many of the player’s chosen numbers match the numbers drawn.

Match all five of the numbers on the white balls and the one on the red powerball, and you win the jackpot. After that, smaller prizes are given out for matching some subset of the numbers.

The Powerball website helpfully provides a list of the odds and prizes for each of the possible outcomes. We can use those probabilities and prize sizes to evaluate the expected value of a $2 ticket.

Take each prize, subtract the price of our ticket, multiply the net return by the probability of winning, and add all those values up to get our expected value:

At first glance, we end up with a positive expected value of $0.72. That seems as though it might make sense to buy a ticket, but considering other aspects of the lottery makes things much worse.

Annuity versus lump sum

Looking at just the headline prize is a vast oversimplification.

First, the $700 million jackpot is paid out as an annuity, meaning that rather than getting the whole amount all at once, it’s spread out in smaller — but still multimillion-dollar — annual payments over 30 years. If you choose instead to take the entire cash prize at one time, you get much less money up front: The cash payout value at the time of writing is $443.3 million.

If we take the lump sum, then, we end up seeing that the expected value of a ticket drops below zero, to -$0.16, suggesting that a ticket for the lump sum is a bad deal:

The question of whether to take the annuity or the cash is somewhat nuanced. The Powerball website says the annuity option’s payments increase by 5% each year, presumably keeping up with and somewhat exceeding inflation.

On the other hand, the state is investing the cash somewhat conservatively, in a mix of US government and agency securities. It’s quite possible, although risky, to get a larger return on the cash sum if it’s invested wisely.

Further, having more money today is frequently better than taking in money over a long period, since a larger investment today will accumulate compound interest more quickly than smaller investments made over time. This is referred to as the time value of money.

Taxes make things much worse

In addition to comparing the annuity with the lump sum, there’s also the big caveat of taxes. While state income taxes vary, it’s possible that combined state, federal, and, in some jurisdictions, local taxes could take as much as half of the money.

Factoring this in, if we’re taking home only half of our potential prizes, our expected-value calculations move deeper into negative territory, making our Powerball investment an increasingly bad idea.

Here’s what we get from taking the annuity, after factoring in our estimated 50% in taxes. The expected value drops to -$0.48:

The tax hit to the lump-sum prize is just as damaging:

Even if you win, you might split the prize

Another potential problem is the possibility of multiple jackpot winners. Bigger pots, especially those that draw significant media coverage, tend to bring in more customers for lottery tickets. And more people buying tickets means a greater chance that two or more will choose the magic numbers, leading to the prize being split equally among all winners.

It should be clear that this would be devastating to the expected value of a ticket. Calculating expected values factoring in the possibility of multiple winners is tricky, since this depends on the number of tickets sold, which we won’t know until after the drawing. However, we saw the effect of cutting the jackpot in half when considering the effect of taxes. Considering the possibility of needing to cut the jackpot in half again, buying a ticket is almost certainly a losing proposition if there’s a good chance we’d need to split the pot.

One thing we can calculate fairly easily is the probability of multiple winners based on the number of tickets sold. The number of jackpot winners in a lottery is a textbook example of abinomial distribution, a formula from basic probability theory. If we repeat some probabilistic process some number of times, and each repetition has some fixed probability of “success” as opposed to “failure,” the binomial distribution tells us how likely we are to have a particular number of successes.

In our case, the process is filling out a lottery ticket, the number of repetitions is the number of tickets sold, and the probability of success is the 1-in-292,201,338 chance of getting a jackpot-winning ticket. Using the binomial distribution, we can find the probability of splitting the jackpot based on the number of tickets sold:

It’s worth noting that the binomial model for the number of winners has an extra assumption: That lottery players are choosing their numbers at random. Of course, not every player will do this, and it’s possible that some numbers are more frequently chosen than others. That would make the odds of splitting the jackpot slightly higher if a more popular number is drawn Wednesday night. Still, the above graph gives us at least a good idea of the chances of a split jackpot.

Most Powerball drawings don’t have too much of a risk of multiple winners — the average in 2017 so far has sold about 22 million tickets, according to our analysis of records from LottoReport.com, leaving only about a 0.3% chance of a split pot.

Larger prizes, however, tend to draw more contenders. Saturday’s drawing, when the jackpot was $535 million, sold about 114 million tickets, according to LottoReport.com . That still leaves only about a 6% chance of two or more winners.

However, in January 2016, when the jackpot topped $1 billion — and eventually $1.5 billion — a whopping 635 million tickets were sold. In that drawing, it would have been surprising if there hadn’t been a split pot, with a nearly 2-in-3 chance of two or more winners. In the end, three people won the jackpot.

With the jackpot now getting closer to that historic high, this week’s drawings could bring in hundreds of millions of customers, increasing the possibility of a split pot.

That leads to a conundrum: Ever huger jackpots, which should lead to a better expected value of a ticket, could have the unintended consequence of bringing in too many new players, increasing the odds of a split jackpot and damaging the value of a ticket.

To anyone still playing the lottery despite all this, good luck!

Even though Wednesday's Powerball has the second-highest jackpot ever, it still doesn't make mathematical sense to buy a ticket.

Powerball expected value

Durango Bill’s
Applied Mathematics

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
——————————————————————–
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.

Probability of not matching anything
Match 0 out of 5 white numbers and not match the Powerball
The number of ways 0 of the 5 first numbers on your lottery ticket can match the 5 white balls is COMBIN(5,0) = 1. The number of ways the 5 losing initial 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 final number can fail to match the Powerball number is: COMBIN(25,1) = 25. The product of these numbers is the number of ways you can get this configuration: COMBIN(5,0) x COMBIN(64,5) x COMBIN(25,1) = 190,612,800. The probability of failing to match anything is thus: 190,612,800/292,201,338 = 0.65233377 or just under two times out of every 3 tickets.

(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, the number of tickets in play is approximately equal to the dollar increase in the annuity Jackpot. For example, if the preceding game had an annuity payout amount of $350,000,000 and the current game has an annuity payout amount of $400,000,000, then there are about 400,000,000 – 350,000,000 = 50,000,000 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.lottoreport.com/ticketcomparison.htm)

The following table gives the probabilities that exactly “K” tickets will share a Jackpot given that there are “N” tickets in play.

“N” Number “K”
of tickets Number of tickets holding the Jackpot combination
in play 0 1 2 3 4 5 6
——————————————————————– –
100,000,000 0.7102 0.2430 0.0416 0.0047 0.0004 0.0000 0.0000
200,000,000 0.5044 0.3452 0.1181 0.0270 0.0046 0.0006 0.0001
300,000,000 0.3582 0.3678 0.1888 0.0646 0.0166 0.0034 0.0006
400,000,000 0.2544 0.3482 0.2383 0.1088 0.0372 0.0102 0.0023
500,000,000 0.1807 0.3091 0.2645 0.1509 0.0645 0.0221 0.0063
600,000,000 0.1283 0.2634 0.2705 0.1851 0.0950 0.0390 0.0134
700,000,000 0.0911 0.2183 0.2615 0.2088 0.1250 0.0599 0.0239
800,000,000 0.0647 0.1772 0.2425 0.2213 0.1515 0.0830 0.0379

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
In Millions 100 200 300 400 500 600 700 800 900 1000
———————————————————————–
100 0.61 0.90 1.19 1.48 1.77 2.06 2.35 2.64 2.93 3.22
200 0.57 0.82 1.06 1.31 1.56 1.81 2.05 2.30 2.55 2.80
300 0.53 0.75 0.96 1.18 1.39 1.60 1.82 2.03 2.25 2.46
400 0.51 0.69 0.88 1.07 1.25 1.44 1.62 1.81 2.00 2.18
500 0.48 0.65 0.81 0.98 1.14 1.30 1.47 1.63 1.79 1.96
600 0.47 0.61 0.76 0.90 1.05 1.19 1.34 1.48 1.63 1.77
700 0.45 0.58 0.71 0.84 0.97 1.10 1.23 1.36 1.49 1.62
800 0.44 0.55 0.67 0.79 0.90 1.02 1.14 1.26 1.37 1.49
900 0.43 0.53 0.64 0.74 0.85 0.96 1.06 1.17 1.27 1.38
1000 0.42 0.51 0.61 0.71 0.80 0.90 1.00 1.09 1.19 1.29

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 Prob. of Expected After
Match Power Play Power Play of result Value Taxes
5 for 5 not PB 1,000,000 1,000,000 8.556E-08 0.0856 0.0513
4 for 5 with PB 50,000 88,372.09 1.095E-06 0.0968 0.0581
4 for 5 not PB 100 176.74 2.738E-05 0.0048 0.0048
3 for 5 with PB 100 176.74 6.899E-05 0.0122 0.0122
3 for 5 not PB 7 12.37 0.0017248 0.0213 0.0213
2 for 5 with PB 7 12.37 0.0014259 0.0176 0.0176
2 for 5 not PB 4 7.07 0.0108722 0.0769 0.0769
1 for 5 with PB 4 7.07 0.0260934 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
———————————————————
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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Powerball odds and probabilities for the Powerball Jackpot. How to calculate these Powerball odds. Jackpot split probabilities including return on investment calculations.