# 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 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

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 + ?

## Jackpot Sharing

The next chart shows the estimated number of tickets sold (in millions), estimated number of winners, and probability of at least one winner by jackpot size (in millions).

### Jackpot Sharing

Jackpot
(Millions)
Tickets
Sold
(Millions)
Estimated
Winners
Probability
1+ Winner
\$40 7.41 0.0254 2.51%
\$50 7.60 0.0260 2.57%
\$60 7.79 0.0267 2.63%
\$70 7.99 0.0274 2.70%
\$80 8.19 0.0280 2.77%
\$90 8.40 0.0288 2.83%
\$100 8.61 0.0295 2.91%
\$120 9.06 0.0310 3.05%
\$140 9.52 0.0326 3.21%
\$160 10.01 0.0343 3.37%
\$180 10.52 0.0360 3.54%
\$200 11.06 0.0379 3.71%
\$250 12.53 0.0429 4.20%
\$300 14.20 0.0486 4.74%
\$350 16.09 0.0551 5.36%
\$400 18.24 0.0624 6.05%
\$450 27.77 0.0950 9.06%
\$500 37.30 0.1276 11.98%
\$550 46.82 0.1602 14.81%
\$600 56.35 0.1929 17.54%
\$650 65.88 0.2255 20.19%
\$700 75.41 0.2581 22.75%
\$750 84.94 0.2907 25.23%
\$800 94.47 0.3233 27.63%
\$850 104.00 0.3559 29.95%
\$900 113.53 0.3885 32.20%
\$950 123.06 0.4211 34.37%
\$1,000 132.59 0.4538 36.48%
\$1,100 151.65 0.5190 40.49%
\$1,200 170.71 0.5842 44.25%
\$1,300 189.77 0.6494 47.77%
\$1,400 208.83 0.7147 51.06%
\$1,500 227.89 0.7799 54.15%
\$1,600 246.94 0.8451 57.05%
\$1,700 266.00 0.9103 59.76%
\$1,800 285.06 0.9756 62.30%
\$1,900 304.12 1.0408 64.68%
\$2,000 323.18 1.1060 66.91%

Estimated number of tickets sold is based on data from 10-7-15 (the date the Powerball changed the rules to 69 white balls and 26 red balls) to 11-7-22 (the date of a \$2.04 billion jackpot). Based on this data, I find demand to have an exponential relationship with jackpot size up to jackpots of about \$400 million. After that, I show the relationship to be linear. It is hard to estimate demand for large jackpots because there have been only 62 jackpots of \$400 million or more and only five of a billion or more at the time of this writing on November 9, 2022. However, following is my best estimate for number of tickets sold (in millions) by jackpot size (j) in millions.

If j <= 400, tickets sold = 6.7092*exp(0.0025*j)
If j > 400, tickets sold = 0.19059*j - 58

## 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

## Double Play

Starting in August 2021 some states launched an additional game down as Double Play. As of November, 2022, states that offer the Double Play option are Colorado, Florida, Indiana, Maryland, Michigan, Missouri, Montana New Jersey, Pennsylvania, Puerto Rico, South Carolina, South Dakota, Tennessee, Washington.

The Double Play is an optional additional \$1 bet on top of a Powerball ticket. Thirty minutes after every Powerball drawing there will be a Double Play drawing, using the same Powerball rules of drawing five balls without replacement from a pool of 69 white balls and one ball from a pool of 26 red balls, known as the Powerball. The same set of numbers chosen for the Powerball drawing will also be used for the Double Play Drawing. If the player purchases both the Power Play and Double Play, the multiplier will NOT apply to the Double Play game.

The following is the pay table for the Double Play drawing according to the number of white balls that match as well as the Powerball.

• Match 5 + Power Ball pays \$10,000,000
• Match 4 + Power Ball pays \$50,000
• Match 3 + Power Ball pays \$500
• Match 2 + Power Ball pays \$20
• Match 1 + Power Ball pays \$10
• Match 0 + Power Ball pays \$7
• Match 5 pays \$500,000
• Match 4 pays \$500
• Match 3 pays \$20

The following table shows my analysis of the Double Pay feature by itself. The lower right cell shows an expected return of 53.43%, which is a higher return, most of the time, compared to the Powerball by itself.

### Double Pay Analysis

White
Balls
Match
Power
Ball
Matches
Win Combinations Probability Return
5 Yes 10,000,000 1 0.000000 0.034223
4 Yes 50,000 320 0.000001 0.054757
3 Yes 500 20,160 0.000069 0.034497
2 Yes 20 416,640 0.001426 0.028517
1 Yes 10 3,176,880 0.010872 0.108722
0 Yes 7 7,624,512 0.026093 0.182653
5 No 500,000 25 0.000000 0.042779
4 No 500 8,000 0.000027 0.013689
3 No 20 504,000 0.001725 0.034497
2 No - 10,416,000 0.035647 0.000000
1 No - 79,422,000 0.271806 0.000000
0 No - 190,612,800 0.652334 0.000000
Total   - 292,201,338 1.000000 0.534334

## 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.

SuperEnalotto € 67,600,000
Euro Millions € 17,000,000
La Primitiva € 14,200,000
Lotto £ 11,400,000
El Gordo € 10,300,000
Oz Lotto AU\$ 10,000,000
Lotto € 6,000,000
Lotto 6/49 CA\$ 9,000,000
Lotto 6aus49 € 4,000,000
Powerball AU\$ 6,000,000
FDJ Loto € 3,000,000
Thunderball £ 500,000
Mega Millions \$ 346,000,000
Lotto \$ 4,400,000
Lotto \$ 4,500,000
Hoosier Lotto \$ 20,000,000
SuperLotto Plus \$ 62,000,000
Hot Lotto \$ 9,100,000
Powerball \$ 307,000,000