@pp99,
The first one is easy. There are four states (0,3), (1,2), (2,1), and (3,0). The first and last are terminal states, and the game starts in state (1,2). The probability of going from (1,2) to (0,3) is 1/3. The probability of going from (1,2) to (2,1) to (1,2) is 2/3 * 1/3 = 2/9. Therefore, the probability of A losing is 1/3 + 1/3*2/9 + 1/3*2/9*2/9 + ...
This is 1/3 * (1 + 2/9 + (2/9)^2 + (2/9)^3 + ...), which is 1/3 times the sum of a geometric series with r=2/9.
Therefore the probability is 1/3 * 1/(1-2/9) = 1/3 * 1/(7/9) = 1/3 * 9/7 = 3/7.