Taking a two-point-margin ending in badminton

Question

A badminton game is tied and must be won by two points. On each rally you win with probability 0.7 and your opponent with probability 0.3, independently. What is the probability you win from the tie?

Accepted Answer

Let w be your win probability from the tie. Over the next two rallies: - Win both, probability 0.7^2 = 0.49: won. - Lose both, probability 0.3^2 = 0.09: lost. - Split, probability 2(0.7)(0.3) = 0.42: tied again. w = 0.49 + 0.42\, w \;\Longrightarrow\; 0.58\, w = 0.49 \;\Longrightarrow\; w = \frac{0.49}{0.58} = \frac{49}{58} \approx 0.845. Takeaway. The win-by-two answer \frac{p^2}{p^2+q^2} here equals \frac{49}{58}, comfortably above the raw point rate of 0.7.

Taking a two-point-margin ending in badminton

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