A and B play a game of tennis. The situation of the game is as follows If one scores two consecutive points after a deuce he wins; if loss of a point is followed by win of a point, it is deuce. The chance of a server to win a point is 2/3. The game is at deuce and A is serving. Probability that A will win the game is, (serves are changed after each point) (a) 3/5 (b) 2/5 (c) 1/2 (d) 4/5 The answer is c) but i am getting D) This is how i am getting D) The probability of getting a deuce is 4/9 so probability that A wins is $$\frac{2}{3})^2[1+4/9.....]$$

The answer is $\frac 12$, because of the symmetry. To note this find a sequence of points in which A wins and then you can easily find a sequence of points with the same probability that wins for B and vice versa.