Expected number of shared-birthday pairs

Question

There are 50 people in a room, with birthdays independent and uniform over 365 days. On average, how many pairs of people share a birthday?

Accepted Answer

"On average, how many pairs" is a job for linearity of expectation, which lets you add up per-pair chances even though the pairs overlap. Count the pairs of people: with 50 people that is \binom{50}{2} = \frac{50 \cdot 49}{2} = 1225 	ext{ pairs}. Each specific pair shares a birthday with probability 1/365 (fix one person's day, the other must match it). Give each pair a 0/1 indicator that equals 1 when they match. The expected number of matching pairs is the sum of these indicator expectations: E[	ext{matching pairs}] = 1225 	imes \frac{1}{365} = \frac{1225}{365} \approx 3.36. Linearity of…

Expected number of shared-birthday pairs

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