Serving out a deuce game in tennis

Question

A tennis game is at deuce, where a player must win by two points. On each point you win with probability 	frac23 and your opponent with probability 	frac13, independently. What is the probability you win the game from deuce?

Accepted Answer

Let w be your win probability from deuce. Look at the next two points: - You win both, probability (	frac23)^2 = 	frac49: game won. - Opponent wins both, probability (	frac13)^2 = 	frac19: game lost. - One each, probability 2 \cdot 	frac23 \cdot 	frac13 = 	frac49: back to deuce. w = 	frac49 + 	frac49 w \;\Longrightarrow\; 	frac59 w = 	frac49 \;\Longrightarrow\; w = \frac45 = 0.8. This is \frac{p^2}{p^2 + q^2} = \frac{4/9}{4/9 + 1/9}. Takeaway. From deuce the win chance is \frac{p^2}{p^2+q^2}; winning two-thirds of points converts to winning four-fifths of deuce games.

Serving out a deuce game in tennis

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