The underdog in a win-by-two tiebreak

Question

A tiebreak is tied and must be won by two points. On each point you win with probability 0.4 and your opponent with probability 0.6, 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 points: - Win both, probability 0.4^2 = 0.16: won. - Lose both, probability 0.6^2 = 0.36: lost. - Split, probability 2(0.4)(0.6) = 0.48: tied again. w = 0.16 + 0.48\, w \;\Longrightarrow\; 0.52\, w = 0.16 \;\Longrightarrow\; w = \frac{0.16}{0.52} = \frac{4}{13} \approx 0.308. Equivalently \frac{p^2}{p^2+q^2} = \frac{0.16}{0.52}. Takeaway. Win-by-two also magnifies a deficit: a 0.4 point rate shrinks to a 4/13 chance. The underdog's odds fall below their point share.

The underdog in a win-by-two tiebreak

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