From Mind Your Decisions, a puzzle about gambling:
Your friend wants to make an even-payoff bet on the outcome of the entire World Series. That is, he wants to make a $100 bet so that if his team is the champion he will win $100, and if his team loses he will lose all of his money.
The problem is he uses a bookie that takes bets only on individual games, and not the entire outcome. The bookie is, however, offering even-payout bets for each game and for any dollar amount.
How much should your friend bet on each game so that he can simulate an even-payout $100 bet on the outcome of the entire series?
For notational simplicity, I’m going to measure money in units of $100, so you start with . And for concreteness, let’s say you want to bet on the Giants against the Royals. (I used to live in San Francisco and have never been anywhere near Kansas City.) The goal is to put together a series of bets that will leave you with if the Giants win and if they lose.
The “probabilities” that I’m going to mention are probabilities computed as if all games are independent and equally likely to be won by both teams; of course this is not true in reality. (The finance folks have a name for this; it’s been a while since I looked at any finance. What is it?)
The answer can be summarized as follows: To determine what to bet on the Giants in game , before game but after game :
- determine the probability that the Giants will win the series if they win game ; call this ;
- determine the probability that the Giants will win the series if they lose game ; call this ;
Now, note that the winning probability before game must be .
By following this strategy, if the Giants win your bankroll goes up by , and the probability of the Giants winning goes up by or ; that is, the change in your bankroll is twice the change in probability. This is also true if the Giants lose. At the beginning your bankroll is and the probability of a Giants win is , so your bankroll is always twice the win probability. In the end it’s 2 if the Giants win and 0 if they lose, simulating the desired bet.
On a related note, people’s guesses about how scores proceed in an NFL game are wrong.