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Last Updated: Jan. 15, 2016

Powerball Lottery


Introduction



The Powerball is a multi-state lottery known for very large jackpots. It started in 1988 under the name Lotto America. In 1992, the name was changed to Powerball. As of this writing, Powerball can be played in 44 states.

Rules



The rules were changed on October 7, 2015 to make the jackpot harder to hit, and thus grow larger jackpots. The previous jackpot probability was 1 in 175.2 million. It is now 1 in 292.2 million. The full rules are as follows:

  1. The cost of a ticket is $2. The player may also pay an extra $1 to enable the Power Play multiplier, except in California.
  2. The player shall pick five distinct numbers from 1 to 69, as well as one "Power Ball" from 1 to 26.
  3. Every Saturday and Wednesday, at 10:59 PM eastern time, the lottery shall draw five white balls numbered 1 to 69 and one red Power Ball from 1 to 26.
  4. The player shall win according to how many of his picks match the ball draw, as shown in the table below.

    Pay Table

    White
    Balls
    Match
    Power
    Ball
    Matches
    Win
    5 Yes Jackpot
    5 No $1,000,000
    4 Yes $10,000
    3 Yes $100
    4 No $100
    2 Yes $7
    3 No $7
    1 Yes $4
    0 Yes $4


  5. If the player invokes the Power Play option, then any prizes other than the jackpot will be multiplied by at least 2. The multiplier for the $1,000,000 prize is always 2. Otherwise, the possible multipliers are shown below.

    Jackpots Under $150 Million

    Multiplier Weight Probability
    10 1 0.023256
    5 2 0.046512
    4 3 0.069767
    3 13 0.302326
    2 24 0.558140
    Total 43 1.000000


    Jackpots Over $150 Million

    Multiplier Weight Probability
    5 2 0.047619
    4 3 0.071429
    3 13 0.309524
    2 24 0.571429
    Total 42 1.000000


  6. The advertised jackpot amounts are paid as a 30-year annuity, with the first payment payable immediately at 1.5051435% of the total amount payable. Future payments will go up by 5% per year, compounded annually.
  7. In lieu of the annuity, the player may opt for a lump sum, which will be about 61% of face value.
  8. The pay table is different for California, because all prizes must be paid on a pari-mutual basis in that state.
  9. As with any lottery with a big progressive, if more than one player wins, then the jackpot will be split evenly among all winners.
  10. As with any lottery with large prizes, if a very frequently chosen set of numbers hits, like the Lost numbers of 4-8-15-16-23-42, then prizes, including fixed ones, may have to be reduced.


California Rules



As stated above, all prizes in California must be paid on a parimutuel basis, meaning all prizes must be progressive in nature. The state withholds approximately 50% of all sales as profit and divides the other 50% as follows between the various prize pools. The lower the jackpot size, the greater its contribution rate will be.

California Prize Allocation

White
Balls
Power
Ball
Contribution
Rate
5 Yes 60.0131% to 68.0131%
5 No 8.5558%
4 Yes 2.1903%
3 Yes 1.0951%
4 No 1.1380%
2 Yes 1.3109%
3 No 1.2405%
1 Yes 5.6536%
0 Yes 10.8027%
0 to 2 No 0.0000%
Reserves 0% to 8%
Total 100.0000%


There is no Power Play option in California.

Source: California Lottery Regulations — See section 3.7 starting on page 39.

Powerball Analysis



The following graph shows the number of Powerball tickets sold (in millions) along the y axis by the jackpot size (in millions) along the x axis. The two black line show my estimate of sales by jackpot size. You can see the relationship is exponential through the jackpot of $529 million of Jan 6, 2016. From that point forward, I show a logarithmic relationship. If anybody has ideas of a single curve to fit the data points in the graph, I'm all ears.

The following table shows the probability and expected return for all possible outcomes, assuming the player did not invoke the Power Play option. The return column is the product of the win, probability and 0.5. The reason for dividing by 2 is the cost of a ticket is $2. The lower right cell shows the player can expect to get back 13.8% of his money in the form of fixed prizes (all wins except the jackpot). It also shows the probability of any win is 4.02%.

Powerball Return Table — Before Annuity, Taxes, and Jackpot Sharing

White
Balls
Match
Power
Ball
Matches
Win Combinations Probability Return
5 Yes Jackpot 1 0.00000000342 ?
5 No $1,000,000 25 0.00000008556 0.04277872266
4 Yes $10,000 320 0.00000109514 0.00547567650
3 Yes $100 20,160 0.00006899352 0.00344967620
4 No $100 8,000 0.00002737838 0.00136891913
2 Yes $7 416,640 0.00142586616 0.00499053156
3 No $7 504,000 0.00172483810 0.00603693334
1 Yes $4 3,176,880 0.01087222948 0.02174445895
0 Yes $4 7,624,512 0.02609335074 0.05218670149
0 to 2 No $0 280,450,800 0.95978615950 0.00000000000
Total 292,201,338 1.00000000000 0.13803161983 + ?


The next chart shows the estimated number of tickets sold, probability of a winner, gross return (before considering the annuity and taxes), and net return (after considering the annuity and taxes). All figures should be considered as estimates. The "net return" assumes that the winner will first lose 39% of the top two prizes due to the annuity and then another 39.6% of what is left due to taxes. This does not consider any state incomes taxes.

Return to Player by Jackpot Size (in Millions)

Jackpot
(Millions)
Tickets
Sold
(Millions)
Probability
Winner
Return
(Gross)
Return
(Net)
$40 9.87 3.32% 20.53% 13.58%
$50 10.43 3.51% 22.21% 14.20%
$60 11.03 3.70% 23.88% 14.81%
$70 11.66 3.91% 25.55% 15.43%
$80 12.33 4.13% 27.21% 16.04%
$90 13.03 4.36% 28.87% 16.65%
$100 13.77 4.60% 30.52% 17.26%
$120 15.39 5.13% 33.81% 18.47%
$140 17.20 5.72% 37.07% 19.67%
$160 19.22 6.37% 40.30% 20.86%
$180 21.48 7.09% 43.50% 22.04%
$200 24.00 7.89% 46.66% 23.21%
$250 31.69 10.28% 54.34% 26.04%
$300 41.83 13.34% 61.63% 28.72%
$350 55.22 17.22% 68.37% 31.21%
$400 72.89 22.08% 74.38% 33.42%
$450 96.23 28.06% 79.41% 35.27%
$500 127.03 35.26% 83.19% 36.67%
$550 198.91 49.37% 82.07% 36.25%
$600 235.44 55.32% 84.30% 37.07%
$650 269.05 60.18% 86.50% 37.88%
$700 300.17 64.20% 88.66% 38.68%
$750 329.14 67.58% 90.80% 39.47%
$800 356.23 70.45% 92.91% 40.25%
$850 381.69 72.92% 94.99% 41.02%
$900 405.69 75.05% 97.05% 41.77%
$950 428.39 76.92% 99.09% 42.52%
$1,000 449.93 78.56% 101.10% 43.27%
$1,100 489.95 81.30% 105.07% 44.73%
$1,200 526.48 83.50% 108.96% 46.16%
$1,300 560.09 85.29% 112.79% 47.57%
$1,400 591.21 86.78% 116.55% 48.96%
$1,500 620.17 88.03% 120.26% 50.32%
$1,600 647.27 89.09% 123.91% 51.67%
$1,700 672.73 90.00% 127.52% 53.00%
$1,800 696.73 90.79% 131.08% 54.31%
$1,900 719.43 91.47% 134.59% 55.61%
$2,000 740.97 92.08% 138.07% 56.89%


Power Play Analysis



As stated in the rules section above, if the jackpot is under $150 Million a 10x ball is added to the Power Play hopper, resulting in an average multiplier of 119/43, or 2.7674%. The following table shows the additional win with the Power Play feature invoked for jackpots under $150 million. The pays column shows the additional win above the standard Powerball win. Remember that the multiplier for the $1,000,000 prize is always 2x. The lower right cell shows a return of 42.23%

Power Play Return Table — Jackpot Under 150 Million

White
Balls
Match
Power
Ball
Matches
Win Combinations Probability Return
5 Yes $0 1 0.0000000034 0.0000000000
5 No $1,000,000.00 25 0.0000000856 0.0855574453
4 Yes $17,674.42 320 0.0000010951 0.0193558797
3 Yes $176.74 20,160 0.0000689935 0.0121942042
4 No $176.74 8,000 0.0000273784 0.0048389699
2 Yes $12.37 416,640 0.0014258662 0.0176409488
3 No $12.37 504,000 0.0017248381 0.0213398574
1 Yes $7.07 3,176,880 0.0108722295 0.0768641340
0 Yes $7.07 7,624,512 0.0260933507 0.1844739215
0 to 2 No $0 280,450,800 0.9597861595 0.0000000000
Total 292,201,338 1.0000000000 0.4222653609


If the jackpot is above $150 million, then the 10x ball is removed from the Power Play hopper, resulting in an average multiplier of 109/42, or 2.5952%. The following table shows the additional win with the Power Play feature invoked for jackpots over $150 million. The lower right cell shows a return of 38.95%.

Power Play Return Table — Jackpot Above 150 Million

White
Balls
Match
Power
Ball
Matches
Win Combinations Probability Return
5 Yes $0 1 0.0000000034 0.0000000000
5 No $1,000,000.00 25 0.0000000856 0.0855574453
4 Yes $15,952.38 320 0.0000010951 0.0174700155
3 Yes $159.52 20,160 0.0000689935 0.0110061098
4 No $159.52 8,000 0.0000273784 0.0043675039
2 Yes $11.17 416,640 0.0014258662 0.0159221721
3 No $11.17 504,000 0.0017248381 0.0192606921
1 Yes $6.38 3,176,880 0.0108722295 0.0693751786
0 Yes $6.38 7,624,512 0.0260933507 0.1665004286
0 to 2 No $0 280,450,800 0.9597861595 0.0000000000
Total 292,201,338 1.0000000000 0.3894595458


Annuity Analysis



Assuming a lump sum offer of 61% and you can invest the lump sum in a way to avoid taxes, then you would need to beat an interest rate of 2.84% to have more more money in the long run taking the lump sum. If you assume a capital gains tax rate of 20%, then you would need to beat 3.92% for the lump sum to be the better value.

Let me also clear up something. There is a pervasive myth that if a lottery winner dies before he receives all his payments, then any future payments will revert back to the lottery/state. The fact is that future payments will be handled in the same way as any other asset -- according to the winner's will. To quote from the Powerball web site:

  • WHAT HAPPENS IF AN ANNUITY PRIZE WINNER DIES?
  • The estate will handle the lottery prize. A lottery annuity prize is just like any other asset. You can pass any remaining annuity payments on to your heirs or to anyone else. The Powerball game will even cash out an annuity prize for an estate. This may make it easier for the estate to distribute the prize. It also may be necessary to cash out the annuity to pay Federal estate taxes. We will sell some or all of the securities at competitive bid or will even just transfer the securities to the estate. We do not charge a fee of any kind. We often hear people complain that the jackpot should not go back to "the state" when a winner dies. It does not. I think that this misunderstanding may come from the response that the prize "goes to the Estate" and some people hear "goes to the State."


Conclusion



Mathematically speaking, the Powerball is an awful bet. After considering the Lottery's cut, jackpot sharing, the annuity, and taxes, the expected return never gets much above 40%. I find the decision to buy lottery tickets to be entirely irrational, and I have yet to understand why people queue up to throw their money away.

Since California legalized the lottery in 1984, where I lived at the time, I have been preaching what an awful bet it is. What is the number of people I have likely saved? As far as I know, zero. Nevertheless, it is my calling to try. I won't go so far as to say that no lottery is ever a good bet. However, I hope I have shown here that the Powerball never is.