Expected shared-birthday pairs in a class

Question

A class has 30 students. Assume each student's birthday is equally likely to be any of 365 days, independently. On average, how many pairs of students share the same birthday?

Accepted Answer

Do not confuse this with "the chance that some pair shares a birthday." Here we want the expected number of matching pairs, which linearity of expectation makes simple. Count the pairs of students: \binom{30}{2} = 435 pairs. For any single pair, the chance they share a birthday is the chance the second person matches the first, P(	ext{a given pair matches}) = \frac{1}{365}. Write the total number of matching pairs as a sum of 435 indicators, one per pair, and add their probabilities: E[	ext{matching pairs}] = 435 	imes \frac{1}{365} = \frac{435}{365} = \frac{87}{73} \approx 1.19. So on ave…

Expected shared-birthday pairs in a class

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