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This is practically the Gambler’s ruin problem with two players involved.We assume that Mr.Adams wins or loses 1$ each game, independent of the past, with probability p, and q respectively, with p+q=1. His objective is to get to a+b dollars before getting ruined.
We let Xn being the fortune after the nth game.
This case Xn yields a Markov chain (MC) on the state space States = {0, 1, : : : ,a+b}.
The transition states also assume transition probabilities, when winning or losing 1$ per game.
Like this: Pa,a+1 =p and Pa,a-1=1-p, because q=1-p....