Winning from deuce in table tennis

Question

A table-tennis game reaches deuce (scores tied, must win by two). On each rally you win the point with probability 0.6 and your opponent with probability 0.4, independently. The game ends when someone leads by two points. What is the probability you win the game from deuce?

Accepted Answer

Let w be your probability of winning starting from deuce. Consider the next two rallies: - You win both (probability 0.6^2 = 0.36): you win the game. - Opponent wins both (probability 0.4^2 = 0.16): you lose. - Split, one each (probability 2(0.6)(0.4) = 0.48): the score is level again, back at deuce. By self-similarity, after a split you are exactly where you began: w = 0.36 + 0.48\, w \;\Longrightarrow\; 0.52\, w = 0.36 \;\Longrightarrow\; w = \frac{0.36}{0.52} = \frac{9}{13} \approx 0.692. Equivalently w = \dfrac{p^2}{p^2 + q^2} with p = 0.6, q = 0.4. Takeaway. A win-by-two race collapses t…

Winning from deuce in table tennis

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