Yesterday my father-in-law asked me to buy him $100 in lottery tickets. He is ordinarily the kind of guy who would cite the quip "the lottery is a tax on people who can't do math," but these are not ordinary times. On Friday night the Mega Millions multi-state lottery will offer a $500 million jackpot, give or take, by far the highest jackpot ever offered in the history of the known universe. The prize is so high it exceeds the number of possible number combinations on a ticket, which is about 176 million. (In other words, the chance that any particular ticket is a winner is about 176 million to one.) The math seems to imply that a $1 ticket has an expected value of $500 million divided by 176 million, or nearly $3. Yet a closer look at the math reveals that the Mega Millions jackpot is a bad bet no matter how large the prize.
The reason? For starters, even if you hit the jackpot, you may have to share it if other ticket holders played the same numbers that you did. Indeed, this is the fate that usually befalls winners of big Mega Millions jackpots. Each of the three previous three highest Mega Millions drawings were won by two different ticket holders who had to split the jackpot; the fourth-highest drawing was split amongst four winning tickets.
Certainly, the threat of having to split is there, but does that really make it a bad bet—especially when the jackpot is so very high? According to the mathematicians, yes. As the number of tickets sold goes up, the chance that more than one person will share in the jackpot does as well, according to a well-known mathematical function called a binomial distribution. When Emory University mathematicians Skip Garibaldi and Aaron Abrams worked through the equations, they found that lotteries are generally a terrible bet—Mega Millions and Powerball particularly so. (I encourage you to take a look at their paper "Finding good bets in the lottery, and why you shouldn't take them," which was published in the American Mathematical Monthly in 2010.)
Even in the case of the current drawing, which offers a jackpot so large that Garibaldi and Abrams show how it should only occur on average every 22 years, the number of tickets that go out is correspondingly large. "I ran the numbers last night," Garibaldi told me over the phone. "You can tell by the amount they estimate the jackpot to be what they estimate the ticket sales to be." Based on the current jackpot, an estimated 380 million tickets have been sold this week. The estimated return on an investment of this week's Mega Millions drawing? Negative 19 percent, per his calculations.
Still, even if that number was positive (and it was, once, in a Texas lottery), buying lottery tickets will never be a sound investment strategy. Using modern portfolio theory, the authors show that the chances of winning the lottery are so incredibly low that the risk precludes any wise investment, no matter what the expected rate of return.
That said, Garibaldi bought five tickets. "I know it's throwing money away," he admitted, but it's fun to be in on the action. That's why I don't feel bad about buying 100 tickets for my father-in-law, either. Besides, he promised to split the jackpot with the rest of the family. With zero out-of-pocket expenses, my rate of return is infinite.
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