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 match is, (serves are changed after each game)

1/2

Let us assume that 'A' wins after n deuces, n ∈ [( 0, ∞)

Probability of a deuce = $\frac{2}{3} . \frac{2}{3}+\frac{1}{3} . \frac{1}{3}=\frac{5}{9}$

(A wins his serve then B wins his serve or A loses his serve then B also loses his serve)

Now probability of 'A' winning the game

$=\sum\limits_{n=0}^{\infty}(5 / 9)^n .\left(\frac{2}{3}\right) \frac{1}{3}=\frac{1}{1-(5 / 9)} . \frac{2}{9}=\frac{1}{2}$