Gamblers A and B have a total of I dollars. They play games of pool repeatedly. Each game they each bet $1, and the winner takes the other’s dollar. The outcomes of the games are statistically independent, and A has probability π and B has probability 1 – π of winning any game. Play stops when one player has all the money. Let Y_t denote A’s monetary total after t games.

State the transition probability matrix. (For this gambler’s ruin problem, 0 and I are absorbing states. Eventually, the chain enters one of these and stays. The other states are transient.)