In How Not to Be Wrong: The Power of Mathematical Thinking
, Jordan Ellenber
g shows us that math isn’t confined to abstract incidents that never occur in real life, but rather touches everything we do–the whole world is shot through with it.
Should you play the lottery?
It’s generally considered canny to say no. The old saying tells us lotteries are a ‘tax on the stupid,’ providing government revenue at the expense of people misguided enough to buy tickets.
The attraction of lotteries is no novelty
The practice dates back to seventeenth-century Genoa, where it seems to have evolved by accident fromt the electoral system. Every six months, two of the city’s governatori were drawn from the members of the Petty Council. Rather than hold an election, Genoa carried out the election by lot, drawing two slips from a pile containing the names of all 120 councilors. The bets become so popular that gamblers started to chafe at having to wait until Election Day for their enjoyable game of chance; and they quickly realized that if they wanted to bet on paper slips drawn from a pile, there was no need for a election at all.
Adam Smith’s approach
Lotteries quickly spread throughout Europe, and from there to North America. But not everyone applauded this development. Adam Smith was a lottery naysayer. In The Wealth of Nations
, he wrote:
That the chance of gain is naturally overvalued, we may learned from the universal success of loteries. The world neither ever saw, nor ever will see, a perfectly fair lottery, or one in which the whole gain compensated the whole loss; because the undertaker could make nothing by it…In a lottery in which no prize exceeded twenty pounds, though in other respects it approached much nearer to a perfectly fair one than the common state lotteries, there would not be the same demand for tickets. In order to have a better chance for some of the great prizes, some people purchase several tickets; and others, small shares in still great number. There is no, however, a more certain proposition in mathematics, than the more tickets you adventure upon, the more likely you are to be a loser. Adventure upon all the tickets in the lottery, and you lose for certain; and the greater the number of your tickets, the nearer you approach to this certainty.
, he wrote:Smith’s conclusion is not correct
Suppose the lottery has 10 million combinations of numbers and just one is a winner. Tickets cost $1 and the jackpot is $6 million.
The person who buys every single ticket spends $10 million and gets $6 million prize; in other words, just as Smith says, this strategy is a certain loser, to the tune of $4 million. The small-time operator who buys a single ticket is better off–at least she has a 1 in 10 million chance of coming out ahead!
But what if you buy two tickets? Then your chance of losing shrinks, though admittedly only from 9,999,999 in 10 million to 9,999,998 in 10 million. Keep buying tickets and your chance of being a loser keeps going down, until the point where you’ve purchased 6 million tickets. In that case, your chance of winning the jackpot, and thus breaking even, is a solid 60%, and there’s only a 40% chance of you ending up a loser.
What Smith’s argument against lotteries is missing is the notion of expected value, the mathematical formalism that captures the intuition Smith is trying to express. It’s not what you expect but the right way to figure out the right price of an object. In this case, the expected value of the lottery ticket is $0.60.