Question: Consider the gambler's ruin problem (Example) in which two gamblers play the game of "heads or tails." Each time a fair coin lands heads up, player A wins $1 from player B, and each time it lands tails up, player B wins $1 from A. Suppose that, initially, player A has a dollars and player B has b dollars. We know that eventually either player A will be ruined in which case B wins the game, or player B will be ruined in which case A wins the game. Let T be the duration of the game. That is, the number of times A and B play until one of them is ruined. Find E(T).

Example: (Gambler's Ruin Problem)

Two gamblers play the game of "heads or tails," in which each time a fair coin lands heads up player A wins $1 from B, and each time it lands tails up, player B wins $1 from A. Suppose that player A initially has a dollars and player B has b dollars. If they continue to play this game successively, what is the probability that

a) A will be ruined;

(b) the game goes forever with nobody winning?