Probability
Probability
Allen Star Batch
Grade 12
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 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)
Step-by-Step Solution
Key Concept: Model the game as a geometric series where each deuce cycle has probability $5/9$ and A wins with probability $4/9$.
Probability of a deuce is $\frac{2}{3} \cdot \frac{2}{3} + \frac{1}{3} \cdot \frac{1}{3} = \frac{5}{9}$ (both serve correctly or both lose serve). Player A wins after $n$ deuces with probability $(5/9)^n \cdot \frac{2}{3} \cdot \frac{2}{3}$. Summing over all deuces: $\sum_{n=0}^{\infty}(5/9)^n \cdot \frac{4}{9} = \frac{4/9}{1-(5/9)} = \frac{1}{2}$.
Correct Answer: 3