World Series Gamble
Published in Brain Teasers
The winner of the World Series is the first team to win four games, so the maximum number of games that could be played is 7. You want to place a $100 bet (even money) on your team to win the whole series. A bookie says he will not take bets on the whole series, but would be willing to let you bet on each individual game. Is there a betting strategy you could put in place to replicate your $100 bet?
Solution:
On games 1 and 2, bet $31.25. On game 3, bet $25.00 if your team is 2-0 or 0-2, and $37.50 if the record is 1-1. On game 4, bet $12.50 if the record is 3-0 or 0-3, and bet $37.50 if the record is 2-1 or 1-2. For game 5 (if you get to it), bet $25 if the record is 3-1 or 1-3, or $50 if the record is 2-2. If there is a game 6, bet $50 on it. If you get to a game 7 (so the record is 3-3), bet $100.
Here is why this works. At game 7, you need to have had $0 profit or loss from the previous games, so you want to bet $100 to replicate your winner-take all bet. Taking a step back, this means that if your team was 3-2, you would have had to lose enough money to get to $0 profit, and win $100 if they won, so your bet would have had to be $50. (Same on the losing side, so you would lose $100 if your team lost and they were 2-3 coming in to the game, or gotten back to $0 if they won to get to 3-3.) The reverse induction works all the way back to the initial bet of $31.25.
Today's brain teaser courtesy of Braingle.com.
