Does anyone on team A share a birthday with team B?

Question

Team A has 20 people and team B has 25 people, with all birthdays independent and uniform over 365 days. What is the probability that at least one person on team A shares a birthday with at least one person on team B? (Matches within a single team do not count.)

Accepted Answer

This is a cross collision: we compare every A person against every B person, but ignore A-with-A and B-with-B matches. Count the cross pairs: each of the 20 people on A pairs with each of the 25 on B, giving 20 	imes 25 = 500 	ext{ cross pairs}. Each cross pair shares a birthday with probability 1/365. Treating the pairs as roughly independent, the chance of no cross match is approximately P(	ext{no cross match}) \approx e^{-500/365} = e^{-1.370} \approx 0.254. Therefore P(	ext{at least one cross match}) \approx 1 - 0.254 \approx 0.746, about 75\%. Equivalently, the expected number of cro…

Does anyone on team A share a birthday with team B?

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