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.
Sean wants to place a $100 bet (even money) on this team to win the whole series. A bookie says he will not take bets on the whole series, but would be willing to let Sean bet on each individual game.
Is there a betting strategy that Sean could put in place to replicate his $100?
Working under the assumption that the bookie will accept bets in increments other than $100, I believe the following strategy works.
Note that if the series is tied 3-3, if Sean's cumulative balance is $0, then simply betting $100 for his team (let's call it team A) will achieve the desired outcome. So our goal is to end up even in the event that the series is tied after six games.
Now consider the situation where one team is ahead 3-2 after five games. The possible outcomes after this game are either one team wins 4-2, in which case Sean's balance needs to be +/- $100, or else the series ends up tied 3-3, in which case his balance needs to be zero. This can be achieved if his prior balance is +/- $50 (depending on which team is ahead), and he places a $50 bet on his team.
Continuing in a similar manner, if the series is 3-1 after four games, the possible outcomes are that the series ends 4-1, or else the series will be 3-2. Given the requirement that Sean's balance is either $100 or $50 after this game, one possibility is that Sean enters a 3-1 situation with a $75 balance and places a $25 bet on his team.
Now, consider the scenario where the series is tied 2-2 after 4 games. The balance requirements after game 5 is played are either +/- $50, which can be achieved by starting with $0 and placing a $50 bet.
Moving to the scenario where the series is 2-1 after three games, the balance requirements are either $75 (if the leading team wins), or $0 (if the team that's behind wins). This is achieved by starting with $37.5 and placing a $37.5 bet.
Continuing in an analogous manner, we can complete Sean's betting table as follows. If the series is:
- (0-0), initial balance $0, bet $31.25 on A
- (1-0), balance $31.25, bet $31.25 on A
- (2-0), balance $62.5, bet $15 on A
- (3-0), balance $87.5, bet $12.5 on A
- (1-1), balance $0, bet $37.5 on A
- (2-1), balance $37.5, bet $37.5 on A
- (3-1), balance $75, bet $25 on A
- (2-2), balance $0, bet $50 on A
- (3-2), balance $50, bet $50 on A
- (3-3), balance $0, bet $100 on A
The betting matrix is symmetrical, so the bets don't change based on which team is ahead.
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Posted by H M
on 2023-03-09 10:37:40 |
Solution
