Suppose a game is played between two players A and B. On each turn of the game, exactly
one of A or B gets a point. Suppose A is better than B and has a probability of 2/3 of getting
a point on each turn of the game. The first person to get two points ahead in the game is the
winner. What is the probability that A wins the game?
Choices
A) 5/9 B) 4/7 C) 2/3 D) 4/5 E) 8/9
Let W0 be the event that Player A wins the game. The goal is to compute P(W0).   Let W1 be the event that Player A wins the game given that she is ahead by one point. Let   W-1 be the event Player A wins the game given that she is behind by one point. Then, the   probability that Player A wins the game is   P(W0) = P(A is ahead by 1 point)P(W1) + P(A is behind by 1 point)P(W-1)   = (2/3)P(W1) + (1/3)P(W-1): (1)   Now,   P(W1) = P(Player A gets a point on the next turn) + P(Player B gets a point on the next turn)P(W0)   = (2/3) + (1/3)P(W0);   and   P(W-1) = P(Player A gets a point on the next turn)P(W0) = (2/3)P(W0):   Substituting these equations into (1) we have   P(W0) = (2/3)[(2/3) + (1/3)P(W0)] + (1/3)(2/3)P(W0) = 4/9 + (4/9)P(W0):   Therefore, P(W0) = (4/9)(9/5) = 4/5   correct answer 4/5
<result>D</result>
Suppose a game is played between two players A and B. On each turn of the game, exactly
one of A or B gets a point. Suppose A is better than B and has a probability of 2/3 of getting
a point on each turn of the game. The first person to get two points ahead in the game is the
winner. What is the probability that A wins the game?