Question

1. Consider the following "Gambler's Ruin" problem. A gambler starts with a certain number of dollar bills between 1 and 5. During each turn of the game, there is a .55 chance that the gambler wil win a dollar, and a .45 chance that the gamble will lose a dollar. The game ends when the gambler has either S0 or S6. Let Xn represent the amount of money that the gambler has after turn n. (a) Give the one-step transition matrix P for the Markov chain {X,,n > 0} (b) Given that the gambler starts with S2, what is the probability that he will end the game with $6? c) Given that the gambler starts with S2, what is the probability that he will, at some point during the game, have exactly 85? (d) Given that the gambler starts with S5, what is the probability that he will end the game with $6? (e) Comment on the relationship between your answers in parts (b), (c), and (d). (f) What is the expected length (number of turns) of the game?

Answer 1

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