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Baseball betting and probablity

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).

Can somebody please explain how did they get value 31.25 using binomial tree.

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    What exactly don't you understand from the answers you have looked at?2012-07-19

2 Answers 2

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In the answers to the previous question they use a binary tree, but no binomials. The tree is the set of states of the series-the number of games won by each team. Each node has a value which we need to find. The problem states that the value of each terminal node (the ones where one team has won the series) is $\pm 100$. At each non-terminal node, you should bet half the difference of its children, so you have that amount of money when you get there. The easy example is if each team has won 3 games. The value of that node must be zero by symmetry, but you want to be $\pm 100$ after the next game as one team will have won the series, so you should bet $100$. The value of the node 3-2 must be 50, as one child (4-2) has value 100 and the other (3-3) has value 0. So at 3-2 you should bet 50. They kept working backwards this way to get the bet at 0-0 and found 31.25.

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    @Willie is right, there is no tree here, binary or otherwise.2012-07-19
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Let me try to draw ASCII art. (I'll do the best 2 out of 3 games case, other wise the drawing becomes very tedious.)

          0:0          /   \        1:0   0:1       /  \   /  \     2:0   1:1   0:2           / \         2:1  1:2 

You want the final tallies of the amount owed to be

           ?          /   \         ?     ?       /  \   /  \     100    ?    -100           / \        100   -100 

To find out how much to bet you need to fill in the above grid.

You want the outcome at 2:1 to be $+100$, and the outcome at 1:2 to be $-100$. The only way for that to happen is that when the series is tied at 1:1, cumulatively you and your uncle don't owe each other anything, and you place a bet of 100. (Your fortunes can only go up or down the same amount after a game. Which means that the tally before a given game must equal the average of the tallies for the two possible outcomes after the given game.) Which means that the outcome at 1:1 must be $0$.

Now, the outcome at 2:0 has to be $+100$, while at 1:1 it is 0. The only way for that to happen is if the total tally at 1:0 is $+50$, and you bet $50$ there. (Similarly, the outcome at 0:1 has to be $-50$, and you bet $50$ there.)

To achieve the outcome at 1:0 and 0:1, the only way is for you to bet $50$ at the start of the series.