Question

Consider the following gamblers ruin problem. A gambler bets $1 on each play of a game. Each time, he has a probability p of winning and probability q = 1 - p of losing the dollar bet. He will continue to play until he goes broke or nets a fortune of T dollars. Let Xn denote the number of dollars possessed by the gambler after the n-th play of the game. Then Xn+1 = {Xn + 1 with probability p; Xn - 1 with probability 1 - p} for 0 < Xn < T. Xn+1 = Xn for Xn = 0 or Xn = T. Xn is a Markov chain. The gambler starts with X0 dollars, where 0 < X0 < T. (a) Construct the (one-step) transition matrix (b) Let T = 3 and p = 0.55. Find the probabilities of winning T dollars when the initial capital of the gambler is 1, ..., T - 1 dollars. (c) For T = 3 and p = 0.45, find the probability of going bankrupt when the initial capital is 1, ..., T - 1 dollars.

          Consider the following gamblers ruin problem. A gambler bets $1 on each play of a game. Each time, he has a probability p of winning and probability q = 1 - p of losing the dollar bet. He will continue to play until he goes broke or nets a fortune of T dollars. Let Xn denote the number of dollars possessed by the gambler after the n-th play of the game. Then Xn+1 = {Xn + 1 with probability p; Xn - 1 with probability 1 - p} for 0 < Xn < T. Xn+1 = Xn for Xn = 0 or Xn = T. Xn is a Markov chain. The gambler starts with X0 dollars, where 0 < X0 < T. (a) Construct the (one-step) transition matrix (b) Let T = 3 and p = 0.55. Find the probabilities of winning T dollars when the initial capital of the gambler is 1, ..., T - 1 dollars. (c) For T = 3 and p = 0.45, find the probability of going bankrupt when the initial capital is 1, ..., T - 1 dollars.

Consider the following gamblers ruin problem. A gambler bets 1 on each play of a game. Each time, he has a probability p of winning and probability q = 1 - p of losing the dollar bet. He will continue to play until he goes broke or nets a fortune of T dollars. Let Xn denote the number of dollars possessed by the gambler after the n-th play of the game. Then Xn+1 = Xn + 1 with probability p; Xn - 1 with probability 1 - p for 0 < Xn < T. Xn+1 = Xn for Xn = 0 or Xn = T. Xn is a Markov chain. The gambler starts with X0 dollars, where 0 < X0 < T. (a) Construct the (one-step) transition matrix (b) Let T = 3 and p = 0.55. Find the probabilities of winning T dollars when the initial capital of the gambler is 1, ..., T - 1 dollars. (c) For T = 3 and p = 0.45, find the probability of going bankrupt when the initial capital is 1, ..., T - 1 dollars.

Added by Emily C.

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