Drunkard's Walk - Leonard Mlodinow [40]
Looking at the world through the lens of mathematical expectation, one often comes upon surprising results. For example, a recent sweepstakes sent through the mail offered a grand prize of $5 million.14 All you had to do to win was mail in your entry. There was no limit on how many times you could enter, but each entry had to be mailed in separately. The sponsors were apparently expecting about 200 million entries, because the fine print said that the chances of winning were 1 in 200 million. Does it pay to enter this kind of “free sweepstakes offer”? Multiplying the probability of winning times the payoff, we find that each entry was worth 1/40 of $1, or 2.5¢—far less than the cost of mailing it in. In fact, the big winner in this contest was the post office, which, if the projections were correct, made nearly $80 million in postage revenue on all the submissions.
Here’s another crazy game. Suppose the state of California made its citizens the following offer: Of all those who pay the dollar or two to enter, most will receive nothing, one person will receive a fortune, and one person will be put to death in a violent manner. Would anyone enroll in that game? People do, and with enthusiasm. It is called the state lottery. And although the state does not advertise it in the manner in which I have described it, that is the way it works in practice. For while one lucky person wins the grand prize in each game, many millions of other contestants drive to and from their local ticket vendors to purchase their tickets, and some die in accidents along the way. Applying statistics from the National Highway Traffic Safety Administration and depending on such assumptions as how far each individual drives, how many tickets he or she buys, and how many people are involved in a typical accident, you find that a reasonable estimate of those fatalities is about one death per game.
State governments tend to ignore arguments about the possible bad effects of lotteries. That’s because, for the most part, they know enough about mathematical expectation to arrange that for each ticket purchased, the expected winnings—the total prize money divided by the number of tickets sold—is less than the cost of the ticket. This generally leaves a tidy difference that can be diverted to state coffers. In 1992, however, some investors in Melbourne, Australia, noticed that the Virginia Lottery violated this principle.15 The lottery involved picking 6 numbers from 1 to 44. Pascal’s triangle, should we find one that goes that far, would show that there are 7,059,052 ways of choosing 6 numbers from a group of 44. The lottery jackpot was $27 million, and with second, third, and fourth prizes included, the pot grew to $27,918,561. The clever investors reasoned, if they bought one ticket with each of the possible 7,059,052 number combinations, the value of those tickets would equal the value of the pot. That made each ticket worth about $27.9 million divided by 7,059,052, or about $3.95. For what price was the state of Virginia, in all its wisdom, selling the tickets? The usual $1.