My wife and I played the Massachusetts PowerBall lottery last month. In one respect it was a good thing that we didn’t win: if we had, it would’ve made my job as a math popularizer that much harder. When a lottery-winner says that playing the lottery is a bad investment strategy, it comes across as hypocritical at best.

One way experts in statistics and probability try to explain why buying a lottery ticket every week is a bad retirement plan is by invoking the concept of expected value. To illustrate the idea, imagine for simplicity a scaled-down version of a lottery — a roulette game with a wheel that has 38 pockets. Suppose that you paid $1 to bet that the ball will wind up in a particular pocket, and that the payoff if you guessed right will be $36. Then in 37 possible worlds you win $0 while in 1 possible world you win $36. Your average payoff over those 38 equally likely parallel worlds is (37x$0+1x$36)/38, or about 95 cents, which is less than what you paid to play. If you like the ambiance of casinos (music, drinks, company) but hate the element of uncertainty, or if the idea of other *you*s in other worlds having different destinies freaks you out, you can fold all thirty-eight worlds into one by placing a bet on each pocket. Then you’ll pay $38 and are sure to win $36, for a net loss of $2. (Seems like a bad idea, but maybe the croupier is cute and you’re hoping your unorthodox betting strategy will make a good conversation-starter.) State-run lotteries are a lot like this roulette game: if you were to bet on every “pocket” (ignoring the fact that there might not be enough money in the economy to enable you to do that), you’d lose big time. And it stands to reason that if buying all possible tickets is a bad idea, so is buying just one per week, or just one *this* time.

But getting back to the casino example: wouldn’t it be nice if the casino paid $40 rather than $36 for a winning bet? Then the strategy of betting on all 38 pockets would give you a profit of $2 instead of a loss of $2. A $2 profit doesn’t sound like much, but the strategy scales up: bet $1000 on every pocket and you’re guaranteed to make a $2000 profit. (Don’t have $38,000 lying around? Find some rich friends to front you the money and promise to split the $2000 profit with them.) But how likely is it that a casino would offer such a good deal?

Amazingly, there is a casino that operates this way. Continue reading →