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In a game where each turn is made up of the roll of a die, player E gets one point when the die is even, and player O gets one point when the die is odd. The first player to accumulate 7 points wins the pot. Suppose the die is fair and the game is interrupted with E leading 4−2.

(a) Denote the probability of the die being even as p, the winning number of points as m, E’s current points as e, and O’s current points as o. In terms of p, m, e and o, what is the general recursive formula that fairly divides the money between E and O?

Based on my inference, I can let $p=0.5$, $m=7, e=4$ and $o =2$. From then, what is the process of coming up with a recursive formula from scratch?

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It is easier to work from $a=m-e$ and $b=m-o$, representing that one player is 3 rounds from winning and the other is 5 rounds from winning. From that, it is clear that $$P(a,b)=p\cdot P(a-1,b)+(1-p)P(a,b-1)$$ with boundary conditions $P(0,k)=1$ and $P(k,0)=0$ for all positive integers $k$.

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