# 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 points 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)

Question : 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 points 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)

(A) 3/5

(B) 2/5

(C) 1/2

(D) 4/5

So, the probability that A wins the game after ‘n’ deuces is $\left( \dfrac{5}{9}\right) ^{n}\times \left( \dfrac{2}{3}\right) \times \left( \dfrac{1}{3}\right)$ [After nth deuce A serves and wins, then B serves and loses]. Therefore, the required probability of ‘ A’ winning the game is
\begin{aligned}\sum ^{\infty }_{n=0}\left( \dfrac{5}{9}\right) ^{n}\times \dfrac{2}{3}\times \dfrac{1}{3}\ \Rightarrow \dfrac{1}{1-\dfrac{5}{9}}\times \dfrac{2}{9}\Rightarrow \dfrac{1}{2}\end{aligned}