申論題

3.(24%) Two players are tossing (possibly biased) coins, on each toss, the probability player 1 wins one cent is p, and the probability player 1 loses one cent is q = 1 - p, where c is the total number of pennies of both players. Define a Markov chain {X_n}, where X_n = j means that player 1 has j cents after the n-th toss. The game continues until one player goes broke (the other player wins).

(c) Let T = {1,... , c - 13} is a finite set of transient states and x_j  is the probability that player 1 wins given T. Write down the systems of equations that x_j need to satisfy. (10 points)