Winning a tennis game from deuce

Question

In tennis, a game at deuce is won by the first player to get two points ahead. If the players split the next two points, the score returns to deuce. A player wins each point with probability p, independently of all other points. What is the probability that she wins the game, starting from deuce? Simplify the expression, and evaluate it for p = 0.6.

Accepted Answer

W = p squared divided by (p squared + (1 minus p) squared). For p = 0.6 that is 0.36 / (0.36 + 0.16) = 9/13, about 0.69. Set up the recursion. Write q = 1 minus p. Consider the next two points from deuce. - She wins both: probability p squared, and she wins the game. - She loses both: probability q squared, and she loses the game. - They split: probability 2pq, and it is deuce again, from which she wins with probability W. So W = p squared + 2pq × W. Solve. W (1 minus 2pq) = p squared, so W = p squared / (1 minus 2pq). Since 1 minus 2pq = p squared + q squared (because (p + q) squared = 1 = p…

Winning a tennis game from deuce

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