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Powerball Lottery

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Introduction

Powerball is a regional lottery that offers notably substantial jackpots, originating in 1988 as Lotto America and rebranded to Powerball in 1992. Currently, it is available in 44 states across the United States.

Buy Lottery Tickets Online

Name Jackpot Buy Tickets
SuperEnalotto € 67,600,000 Tickets
Eurojackpot € 30,000,000 Tickets
Euro Millions € 17,000,000 Tickets
La Primitiva € 14,200,000 Tickets
Lotto £ 11,400,000 Tickets
El Gordo € 10,300,000 Tickets
Lotto € 6,000,000 Tickets
Lotto 6/49 CA$ 9,000,000 Tickets
Oz Lotto AU$ 10,000,000 Tickets
Lotto 6aus49 € 4,000,000 Tickets
Powerball NZ$ 7,000,000 Tickets
Powerball AU$ 6,000,000 Tickets
FDJ Loto € 3,000,000 Tickets
Saturday Lotto AU$ 4,000,000 Tickets
Ontario 49 CA$ 2,000,000 Tickets
Daily Million 2PM € 1,000,000 Tickets
Daily Million 9PM € 1,000,000 Tickets
Powerball R 19,500,000 Tickets
Thunderball £ 500,000 Tickets
Monday & Wednesday Lotto AU$ 1,000,000 Tickets
Megabucks $ 5,700,000 Tickets
Lotto Texas $ 6,250,000 Tickets
Fantasy 5 $ 75,000 Tickets
Powerball $ 307,000,000 Tickets
Jersey Cash 5 $ 75,000 Tickets
Hot Lotto $ 9,100,000 Tickets
Two Step $ 575,000 Tickets
SuperLotto Plus $ 62,000,000 Tickets
Hoosier Lotto $ 20,000,000 Tickets
Lotto $ 4,500,000 Tickets
Pick 6 $ 11,900,000 Tickets
Lotto $ 4,400,000 Tickets
Mega Millions $ 346,000,000 Tickets

Rules

As of October 7, 2015, modifications were made to the gameplay to increase the difficulty of winning the jackpot, resulting in larger possible winnings. Previously, the odds were 1 in 175.2 million, but they have been adjusted to 1 in 292.2 million. Detailed rules are outlined below:

  1. A single ticket costs $2, and players can opt to pay an additional $1 for the Power Play feature, although this is not available in California.
  2. Participants must select five unique numbers from the range of 1 to 69, plus one 'Power Ball' number from 1 to 26.
  3. Every Wednesday and Saturday at 10:59 PM Eastern Time, the lottery will conduct a draw involving five white balls numbered from 1 to 69 and one red Power Ball from 1 to 26.
  4. Winners will be determined by the number of winning combinations in relation to the drawn numbers, as illustrated 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 players choose the Power Play option, any winnings excluding the jackpot will be multiplied by at least two. The multiplier for the $1,000,000 prize is consistently set at 2. Other potential multipliers are listed 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 announced jackpot values are disbursed as a 30-year annuity, with an initial payout immediately issued as 1.5051435% of the total jackpot figure. Subsequent payments will increase by 5% annually, compounded over the years.
  7. Instead of receiving payments over time, winners have the alternative to take a lump sum, which typically represents around 61% of the advertised jackpot amount.
  8. The pay table is different for California This policy is enforced because all prizes must be distributed on a pari-mutuel basis in that particular state.
  9. As is common with lotteries offering significant progressive amounts, if multiple winners come forward, the jackpot will be divided equally among those individuals.
  10. In any lottery featuring large rewards, should a popular set of numbers win—such as 4-8-15-16-23-42—then it’s possible the prizes may need to be reduced, including fixed amounts. Lost As previously mentioned, all prizes in California must follow a parimutuel structure, ensuring all payouts are progressive. The state retains roughly 50% of total sales profits, sharing the remaining 50% among various prize pools. A larger contribution rate applies to smaller jackpots.

California Rules

California does not offer the Power Play option.

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%

The subsequent table outlines the odds and expected returns for various outcomes, assuming the Power Play option wasn't chosen. The return column reflects the win amount multiplied by the probability and then divided by 2, as each ticket costs $2. In total, players can anticipate recovering about 13.8% of their expenditure through fixed prize winnings (excluding the jackpot). The overall chance of winning something is stated to be 4.02%.

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

Powerball Analysis

The next graph displays the predicted number of tickets sold (in millions), estimated winners, and the likelihood of having at least one winner based on different jackpot sizes (in millions).

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

Estimates for ticket sales stem from data collected between October 7, 2015 (the date of the rule changes that increased the number of white balls to 69 and red balls to 26) and November 7, 2022 (the date of a $2.04 billion jackpot). Analysis suggests a rising demand correlates exponentially with jackpot sizes up to approximately $400 million. Beyond this threshold, demand appears to follow a linear trend. Estimating the ticket sales for these larger jackpots is challenging due to the limited number of jackpots exceeding $400 million (only 62) and merely five reaching a billion or more as of November 9, 2022. Below is my rough estimate of ticket sales (in millions) based on jackpot size (j) measured 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%  

According to the rules mentioned earlier, if the jackpot is below $150 million, a 10x multiplier ball is introduced into the Power Play selection, yielding an average multiplier of 119/43, or approximately 2.7674%. The following table provides details on additional winnings when the Power Play is activated for jackpots under $150 million. Notably, the $1,000,000 prize consistently has a multiplier of 2x, and the return in the lower right corner is indicated as 42.23%.

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

Power Play Analysis

Power Play Return Overview — Jackpot Under 150 Million

For jackpots exceeding $150 million, the 10x multiplier ball is omitted from the Power Play selection, which leads to an average multiplier of 109/42, or approximately 2.5952%. The table that follows illustrates the added winnings when the Power Play feature is applied to jackpots over $150 million, with the lower right cell indicating a return of 38.95%.

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

Power Play Return Overview — Jackpot Above 150 Million

Beginning in August 2021, several states initiated an additional game called Double Play. As of November 2022, the states that allow players to choose the Double Play feature include Colorado, Florida, Indiana, Maryland, Michigan, Missouri, Montana, New Jersey, Pennsylvania, Puerto Rico, South Carolina, South Dakota, Tennessee, and Washington.

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

Double Play is an optional additional $1 wager made in conjunction with a Powerball ticket. A Double Play drawing occurs 30 minutes after every Powerball draw, using identical rules: drawing five balls without replacement from a pool of 69 white balls and one ball from a pool of 26 red balls, termed the Powerball. The number set selected for the Powerball is also valid for the Double Play drawing. However, if players select both Power Play and Double Play, the multiplier does not apply to the latter.

The following table lists the potential payouts for the Double Play drawing, based on how many white balls match, along with the Powerball.

The next table contains my evaluation of the Double Play feature. The lower right corner indicates an expected return rate of 53.43%, frequently offering higher returns compared to the Powerball alone.

  • 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

It's important to clarify a common misconception. There's a widespread belief that if a lottery winner dies before receiving all their payments, the remaining funds revert to the state lottery. In reality, future payments are managed like any other asset, according to the deceased's last will. Referring to the Powerball website:

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

powerball onlineAssuming 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.

WHAT HAPPENS IF AN ANNUITY PRIZE WINNER DIES?

  • The estate will manage any lottery winnings. A lottery annuity is treated like other property. Remaining annuity payments can be transferred to heirs or other individuals. The Powerball game can also liquidate an annuity prize for the estate, facilitating distribution and possibly addressing federal estate tax obligations. They may sell all or part of the securities at a competitive rate or transfer them to the estate without imposing any fees. Misunderstandings arise when people hear that the prize goes to the estate and misinterpret it as going back to the state.
  • From a mathematical standpoint, participating in the Powerball is a poor financial decision. When considering the lottery's share, shared jackpots, annuity structures, and taxes, the expected payout remains below 40%. Personally, I find the choice to buy lottery tickets to be completely illogical, and I have yet to grasp the reasons behind the enthusiasm with which people line up to spend their money.

Conclusion

Since I witnessed California legalizing the lottery in 1984, during my residence there, I have consistently voiced my opinion on the unlikelihood of it being a sound investment. How many individuals have I probably saved from loss? To my knowledge, none. Nevertheless, I feel it's my mission to continue spreading this awareness. While I won't claim that no lottery could ever be a profitable venture, I hope my insights illustrate that the Powerball certainly isn't worthwhile.

- Is there a correlation between states offering better lottery returns and higher losses per resident compared to those with less favorable lottery outcomes?

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