Is Powerball a Mug's Game?
It all depends on when you play—and what value you put on a dollar.
Wednesday’s $500 million (and growing) Powerball jackpot will have ticket holders in 42 states holding their breath. If you buy a lottery ticket this week in hope of hitting it big, are you a foolish romantic or a canny statistician? Back in 2001, mathematician Jordan Ellenberg used pencil and paper to defend his own choice to buy tickets for a $280 million Powerball drawing. His original 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
(1/80 million) x $280 million + (79,999,999/80 million) x 0 = $3.50.
That's a good bet!"
The problem with that argument is that we weren't playing for the $280 million. We were playing for our share of the $280 million, thanks to the possibility of multiple winners. If I win the Powerball, the chance is pretty good that somebody else is going to win, too. Already my jackpot's down to $140 million. And the more people who play, the more the prize will tend to divvy up. If, as happened in real life this week, four people win, you're looking at just $70 million.
So how many people could you expect to share the prize with? I worked this out (requires Adobe Acrobat). The chance of no winner is about 8 percent; of one winner, 21 percent; of two, 26 percent; of three, 21 percent; of four, 13 percent; of five or more, 9 percent. So Saturday's fourfold victory was a bit of a surprise but not a real shocker. A Powerball enthusiast suggested to me that people are especially fond of picking lucky 21 on the red ball and that this could explain the large number of winners. But in fact, Powerball records show that the number of people who picked 21 on the red ball was only half a percentage point greater than would have been expected by chance. That's not enough to make a serious difference.
Jordan Ellenberg is a professor of mathematics at the University of Wisconsin. His book How Not To Be Wrong is forthcoming. He blogs at Quomodocumque.