`Aa n dB` 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 a deuce and `A` is serving. Probability that `A` will win the match is (serves are change after each game) `3//5` b. `2//5` c. `1//2` d. `4//5`

Options

A. `3//5`
B. `2//5`
C. `1//2`
D. `4//5`

Answer

C

Explanation

Let us assumed that A wins after n decues, `n=0,1,2,3,...` The probability of a deuce is `(2//3)xx(2//3)+(1//3)xx(1//3)=(5//9).` [A wins his serve, then B wins his serve or A losses his serve.] So, the probability that A wins game after n deuces is `(5//9)^(n)xx(2//3)xx(1//3).` [After nth deuce, A serves and wins, then B serves and loses.] Therefore, the required probability of 'A' winning the game is
`underset(n-0)overset(oo)sum((5)/(9))^(n)xx2/3xx1/3=(1)/(1-(5)/(9))xx2/9=1/2`