25 January 2016

Powerball lottery - Odds, expected value, and billion dollar jackpots

Basic odds and expected values

Wikipedia has a good introductory page about lottery mathematics. In short, the number of possibilities for drawing 5 white balls among 69 and one red ball among 26 is C(69,5) * C(26,1) = 292M. Despite these very low odds, 14 people won the jackpot in 2015, and 72 from 2011 to 2015. The chances of winning prizes are listed in the table below. The expected value column is the product of a prize by its odds.

Match Prize Value Odds Expected value
5 whites + red Jackpot $40M to $1.5B 1 in 292M = .00000034% $0.14 to $5.14
5 whites Match-5 $1M 1 in 11.7M = .0000085% $0.09
1-4 whites + red
or 3-4 whites
Consolation $4 to $50k 1 in 24.7 = 4.04% $0.24

The Powerplay mutliplier applies only to consolation prizes. Taking into account the probability of each multiplier, their expected value becomes $0.65. Power Play also doubles the Match-5 prize by 2, so the Match-5 expected value becomes $0.17. While Power Play does increase values, it does not increase them by enough to cover the $1 cost of the Power Play option. Power Play is not worth it.

Huge jackpots do not justify buying a ticket

Intuitively, the January 16 2015 jackpot of $1.5B is so huge that it may seem statistically worth it to buy a ticket: you spend $2 to receive $1.5B / 292M = $5.14. Each ticket would net $3.14. Put another way, one could earn $1.5B after investing 292M * $2 = $584M. The theory is promising, but it has flaws.

First, 635M tickets were sold. An alternative way to find this number is to mutliply the number of winners by the odds of winning: 26.11M * 24.87 = 650M tickets. Based on the method described here, there is an 89% chance that there is another winning ticket. More specifically, the chances of having exactly 0, 1, 2, 3, 4, and 5 winners among the 650M tickets already sold are 11, 25, 27, 19, 11, and 5%. Taking these probabilities into account, $5.14 becomes $5.14 * (11% / 1 + 25% / 2 + 27% / 3 + ... ) = $2.09. So, buying a Powerball ticket is still (barely) worth it. Second, there is a 40% federal income tax on lottery winnings. That brings it down to $1.26 per $2 ticket. Buying a ticket for the $1.5B jackpot is not statistically worth it anymore.

Last, all tickets have to be bought in person. Since Powerball drawings happen twice a week, one would have to buy 70M tickets per day, or 845 per second. CNN reports that in February 1992, an Australian consortium tried to corner a $24 million Virginia Lotto jackpot. But the group was only able to purchase 2.4 million of the 7 million combinations before time ran out.

Billion-dollar jackpots every year

Could the jackpot get big enough that buying a ticket will be statistically worth it? There is no data point after $1.5B, so we can't say. But we can compute the expected time until such a jackpot appears again.

First, For the jackpot to reach $1.5B after 20 drawings, all 957 million tickets from the previous 19 drawings had to be losers. That is a 4% chance. Then, we can adapt the coin-toss method, using ticket sales and jackpot values for the 20 drawings leading to the $1.5B jackpot, to compute the expected time until the jackpot reaches $1.5B. Unless my math is wrong, I found that the expected number of drawings to reach 19 consecutive drawings without winner is 421. With two drawings a week, that means a $1.5B jackpot should appear every 4 years. A billion-dollar jackpot requires 18 consecutive drawings without winner, which can be expected every year.

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