Gamblers Ruin a man has $2, which he is going to bet $1 at a time until he either loses

Chapter 10, Problem 23

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Gamblers Ruin a man has $2, which he is going to bet $1 at a time until he either loses all his money or he wins $5. Assume he wins with a probability of 0.45 and he loses with a probability of 0.55. (a) Construct the transition matrix for this game. (b) On the average, how many times will the process be in nonabsorbing states? (c) What is the expected length of the game? (d) What is the probability the man loses all his money or wins $5? Hint: The fundamental matrix T of the transition matrix P is T = D 1.584283 1.062332 0.635282 0.285877 1.298406 2.360738 1.411737 0.635282 0.949001 1.725456 2.360738 1.062332 0.52195 0.949001 1.298406 1.584283T