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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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marked as duplicate by Henning Makholm, joriki, Did, Jason DeVito, Zev Chonoles Jul 19 '12 at 15:29

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Who is the "they" who got the value 31.25? – Henning Makholm Jul 19 '12 at 11:24
hamna, please don't reask questions that have already been asked here. They will just get closed as duplicates. If you don't understand part of an answer to a question that's been answered, you can ask a new question, linking to the previous one and explaining specifically what part of the answer you don't understand. Just copying the question without linking to the earlier one wastes everyone's time. – joriki Jul 19 '12 at 11:49
There is no mention of binomial trees among the answers to the previous question (of which this is an exact duplicate). What leads to you bring binomials into the picture? – Henning Makholm Jul 19 '12 at 11:50
(It is true enough that the answer happens to be something like $1-2^{-(n+1)}\binom{2n-3}{n-1}$ times the desired final score, but that connection was not made explicitly made in the previous question. Where did you get it?) – Henning Makholm Jul 19 '12 at 11:54
Is there anything specific about Did's answer (in the thread you linked to) that you don't understand? – Willie Wong Jul 19 '12 at 12:56

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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Actually, I think (using my absurd mind reading powers) the answers do not use trees at all! I think in the answers, how one arrives at the $2:2$ node is not important, so the $2:2$ node connects up to $1:2$ and $2:1$ while it connects down to $3:2$ and $2:3$. The presense of closed loops thus make it not a tree. – Willie Wong Jul 19 '12 at 14:50
@Willie is right, there is no tree here, binary or otherwise. – Did Jul 19 '12 at 15:25

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.)

         /   \
       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.

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