Question

Gambler’s Ruin.A gambler, player A, plays a sequence of games against an opponent, player B. In each game, the probability of player A winning is p. If player A wins, he wins $1 which is paid by player B. If he loses a hand with probability q = 1-­‐p, he must pay $1 to player B. The game ends either player B wins all the money from player A, and he is “ruined,” or when player A wins all the money from player B, and she is “ruined.” If we say that the total pot is $N, we can model the game from player A’s perspective, that is, if player A has $k, then player B has $N-­‐k.Let y[k] = the probability of player A winning the game, given that he currently has a bankroll of $k.First-­‐step Analysis:The game can be modeled as a random walk, where:y[k] = p y[k+1] + q y[k-­‐1]for 1≤푘≤푁−1Boundary Conditions:y[0] = 0, player A is ruined, and B winsy[N] = 1, player B is ruined, and A winsa)Find y[k], the probability ofplayer A winning if 푝≠푞?b)Find y[k], the probability of player A winning if 푝=푞=0.5c)Suppose both players startwith the same amount of money, k = N-­‐k. For p = 0.49,find y[k] fora.N=20b.N=100c.N=200d)Comment on what the results imply from c) above.

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