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Problem 1 . 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 with probability p xn+l Xn 1 with probability 1 p xn+l xn for xn=o or Xn = T for O < xn < T Xn is a Markov chain. The gambler starts with Xo dollars, where 0 < Xo < T. (a) Construct the (one-step) transition matrix (b) Let T 3 and p = 0.6. 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 0.4, find the probability of going bankrupt when the initial capital is 1, T 1 dollars.