Is Powerball a Mug's Game?
It all depends on when you play—and what value you put on a dollar.
This evening, lottery ticket holders across the country will tune in for the results of a Mega Millions drawing for an estimated jackpot of $237 million. In 2001, mathematician Jordan Ellenberg explained why buying a lottery ticket isn't always a fool's game. The article is reprinted below.
Were you stupid not to play?
I don't have to ask myself; I played. My father and his Ph.D. in statistics put me in for a 20 percent share of his four tickets. But I got enough razzing from friends and neighbors that I thought it was worth explaining why, from a mathematician's point of view, last Saturday's drawing wasn't necessarily dumb.
The question to ask is: What is the expected value of a lottery ticket? If the expected value is more than a dollar, and the ticket costs a dollar, you should buy a ticket. If the expected value is less than a dollar, you should keep your money.
"Expected value" doesn't just mean "what do you expect?" After all, you probably expect the ticket to be worth nothing. Yet people don't think lottery tickets are worthless; if they did, they wouldn't buy them. "Expected value" as I mean it here is a mathematical definition that assigns a fixed value to an object whose true value is subject to uncertainty.
Suppose an object might be worth either V1 or V2 dollars, and suppose the probability is P1 that it is worth V1, and P2 that it is worth V2. Then the expected value is defined to be
P1 x V1 + P2 x V2.
For instance, suppose you place a bet on a horse that has a 1/10 chance of winning, and the bet pays $100. Then the probability is (1/10) that your ticket will be worth $100 and (9/10) that your ticket will be worth nothing. So, the expected value of the ticket is
(1/10) x $100 + (9/10) x 0 = $10.
Why is $10 a good definition of the value of the ticket? Because if you spent a week at the track and bought, say, 250 such tickets, you'd probably end up winning about 25 times; you'd make $2,500, or $10 per ticket. So, if you were paying more than $10 for each ticket, you'd be a loser; less, and you'd be a winner.
So, what's the expected value of a Powerball ticket? Here's a wrong argument I heard a lot. People who knew the jackpot odds figured: "I've got a 1 in 80 million chance at $280 million, so the expected value is
Jordan Ellenberg is an associate professor of mathematics at the University of Wisconsin. His first novel is The Grasshopper King. He blogs at Quomodocumque.