. In what is called a gambler's ruin problem, consider a gambler who with capital a agrees to play a series of games with an adversary having a capital b(a + b =

c, the total capital). The probability of the gambler winning one game (and with it, one unit of money) is p, and that of losing one unit is q = 1 - p. (There is no draw.) Suppose that successive games are independent. If X„ is the gambler's fortune at time n (at time of the nth game), show that { X„ ,, n = 0, 1,2 ,... ] is a Markov chain. Write down its TPM. Is the chain irreducible? Examine the nature of the states of the chain.