Baseball betting and probablity Here is a question that came up during class discussions on Friday: Your favorite baseball team is playing against your uncle's favorite team in the World Series. At the beginning of each game, you and your uncle bet on the game's outcome. Your uncle, being wealthy and carefree, always lets you choose the amount of the bet. If you bet b dollars and your team wins the game, your uncle gives you an IOU for b dollars. But if they lose the game, you give him an IOU for b dollars. When the series is over, all outstanding IOUs are settled in cash. You would like to walk away with 100 in cash if your team wins the series, and lose 100 if your team loses the series. How much should you bet on the opening game? (For non-baseball fans, the first team to win a total of four games wins the series). I am thinking to apply probability. For example, knowing 100 is the result, moving backward but not exactly sure how to proceed.

31.25.

As you said, draw the graph of the possible scores, linking a given score to the two possible scores after one more game is played. Place a +100 tag on every vertex where you won (these are 4-0, 4-1, 4-2 and 4-3) and a -100 tag on every vertex where you lost (these are 0-4, 1-4, 2-4 and 3-4) and compute the tags of the other vertices backwards.

For example, 3-3 gives 4-3 and 3-4 hence the tag of the vertex 3-3 should be half the sum of the tags of the vertices 4-3 (+100) and 3-4 (-100), that is, 0. Likewise, the tag of the vertex 3-2 should be half the sum of the tags of the vertices 4-2 (which you knew from the beginning to be +100) and 3-3 (which you just computed to be 0), that is, +50. And so on, crawling back until you reach the vertex 0-0 at the beginning of the graph.

Your bet on the opening game should be the tag of the vertex 1-0 and the opposite of the tag of the vertex 0-1. This is 31.25.