Deciding a tied volleyball set by two

Question

A volleyball set is tied late and must be won by two points. On each rally your team wins the point with probability $0.75$ and the other team with probability $0.25$ , independently.

What is the probability your team wins the set from the tie?

Accepted Answer

Let w be your win probability from the tie. Over the next two rallies: - Win both, probability 0.75^2 = 0.5625: set won. - Lose both, probability 0.25^2 = 0.0625: set lost. - Split, probability 2(0.75)(0.25) = 0.375: tied again. w = 0.5625 + 0.375\, w \;\Longrightarrow\; 0.625\, w = 0.5625 \;\Longrightarrow\; w = \frac{0.5625}{0.625} = 0.9. Equivalently \frac{p^2}{p^2+q^2} = \frac{0.5625}{0.625}. Takeaway. Winning by two magnifies an edge: a 0.75 point rate becomes a 0.9 chance of taking the tied set.

Deciding a tied volleyball set by two

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