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Question: A and B play a game of tennis. The situation of the game is as follows; if one scores two consecutiv...

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)

A

3/5

B

2/5

C

½

D

4/5

Answer

½

Explanation

Solution

Let us assume that 'A' wins after n deuces, n ∈ [(0, ∞)
Probability of a deuce = 23⋅23+13⋅13=59\frac { 2 } { 3 } \cdot \frac { 2 } { 3 } + \frac { 1 } { 3 } \cdot \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

= ∑n=0∞(5/9)n⋅(23)13=11−(5/9)⋅29=12\sum _ { n = 0 } ^ { \infty } ( 5 / 9 ) ^ { n } \cdot \left( \frac { 2 } { 3 } \right) \frac { 1 } { 3 } = \frac { 1 } { 1 - ( 5 / 9 ) } \cdot \frac { 2 } { 9 } = \frac { 1 } { 2 }