<p>\(A\) 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)</p>
Step-by-Step Solution
Key Concept: At deuce, the game has a recursive structure: from deuce, A can win 2 consecutive points (probability 4/9), lose 2 consecutive points (probability 1/9), or return to deuce (probability 4/9). Set up an equation P = 4/9 + (4/9)P to find the probability A wins from deuce.
<p><strong>Step 1: Identify probabilities from deuce position</strong></p><p>When A serves from deuce, A wins a point with probability 2/3 and loses with probability 1/3.</p><p><strong>Step 2: Enumerate outcomes from deuce</strong></p><p>Let P(A) = probability A wins from deuce when A serves.</p><p>From deuce:</p><ul><li>A wins 2 consecutive points: (2/3)(2/3) = 4/9 → A wins the match</li><li>A loses 2 consecutive points: (1/3)(1/3) = 1/9 → B wins the match</li><li>A wins then loses: (2/3)(1/3) = 2/9 → Back to deuce (B now serves)</li><li>A loses then wins: (1/3)(2/3) = 2/9 → Back to deuce (B now serves)</li></ul><p><strong>Step 3: Account for serve change</strong></p><p>When the game returns to deuce after A's two points, B serves next. Since serves alternate and B becomes server at deuce, the probability A wins from that deuce is [by symmetry] 1 - P(B wins from deuce with B serving) = 1 - (4/9 + (4/9)·[1-P(A)]).</p><p>However, by the recursive nature with alternating serves, from deuce: P(A) = 4/9 + (4/9)P(A)</p><p><strong>Step 4: Solve the recursive equation</strong></p><p>P(A) = 4/9 + (4/9)P(A)</p><p>P(A) - (4/9)P(A) = 4/9</p><p>(5/9)P(A) = 4/9</p><p>P(A) = 4/5</p><p>∴ Answer: A</p>
Correct Answer: A