Expected number of shared-birthday pairs

Question

30 people are in a room, each with a birthday uniform over 365 days (ignore leap years, assume independence). What is the expected number of pairs of people who share a birthday?

Accepted Answer

Let X be the number of matching pairs. For each of the \binom{30}{2} pairs, define an indicator I{ij} = 1 if persons i and j share a birthday. Then X = \sum{i<j} I{ij}. For any single pair, the second person matches the first with probability 	frac{1}{365}, so E[I{ij}] = 	frac{1}{365}. By linearity of expectation (no independence needed): E[X] = \binom{30}{2}\cdot\frac{1}{365} = \frac{435}{365} \approx 1.19. So on average about 1.19 pairs share a birthday among 30 people. This is why the "birthday problem" tips over near 23 people: the expected number of matching pairs crosses meaningful si…

Expected number of shared-birthday pairs

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