Question: Two players A and B own between them the amount of N dollars and play for unit stakes with each other with the agreement that

Two players A and B own between them the amount of N dollars and play for unit stakes with each other with the agreement that every time a player loses his last dollar, his adversary immediately returns it (the capitals of the players would remain 1 and N 1 after such a play), so that the game can continue forever. In each play,A wins with prob p,0

(a) Model the situation of player A as a MC and find the transition matrix P. (b) Explain why the chain is irreducible, aperiodic and positive.

i. What is the limit of Pn? Explain.

ii. How many times that A holds $3 capital in 100 games in the long run if

p = 1/2?

iii. What is the average number of games between occurrence of that A has

capital $1 when p = 1/2?

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