How to win the lottery

A hundred numbered balls are placed in an urn, and one winning ball will be drawn. People may buy $1 tickets betting on the ball that will be drawn. After the state, which runs the lottery, takes its cut, the lucky ticketholder wins the remaining jackpot. The jackpot is split if there are multiple people who bought the same winning ticket.

Suppose you and 45 other people enter the lottery. But the 45 other people are stupid, and all of them buy the same ticket for the same number, say lucky number 7. You buy 100 tickets, buying every number. What is your expected payoff? Say the state's cut is 30%. The lottery revenues are $145. After subtracting the state's cut, the jackpot is $101.50. If the 7 ball is not chosen, which happens 99% of the time, then you win the entire jackpot. If the 7 ball is chosen the $101.50 jackpot is split 46 ways, so your share is about $2.21. Thus your expected net payoff is -100 + 0.99*101.50 + 0.01*101.50/46 = $0.50706521739 which is positive. Normally any gambling game is a losing proposition (the house always wins), but in this case, the 45 other "stupid people" in the game are essentially offsetting the state's cut.

The next question is, how can one tell if a lottery has positive expected return? In other words, how can one tell if all the other players in the game have bought such a skewed distribution of tickets so as to offset the state's cut? Here are two ways.

If all players choose their tickets randomly, then we should expect a certain distribution of how high the jackpots will rise until they are won. By observing how the jackpots rise compared to how they should theoretically behave for random players, one can obtain a measure of the skew on how people choose their lottery numbers. One may also detect if someone else is also buying all the tickets.

One may obtain a more detailed picture of the numbers people choose by observing the winning numbers each time the jackpot is won. The winning numbers represent an unbiased (?) sampling of all the numbers people choose. Sometimes the lottery also releases information about how many ways the jackpot is split. We can also gain negative information about the numbers people do not pick by observing the winning numbers on days no one wins the jackpot.