How many distinct birthdays among a hundred people?

Question

A group of 100 people have birthdays independent and uniform over 365 days. On average, how many distinct calendar days are somebody's birthday?

Accepted Answer

Trying to track which specific days are used is hard, but linearity of expectation makes it easy: focus on one day at a time and add up. Pick any single day. It is not anyone's birthday only if all 100 people miss it: P(	ext{day is empty}) = \left(\frac{364}{365}ight)^{100} \approx 0.760. So the chance that day is used is 1 - 0.760 = 0.240. Give each of the 365 days an indicator that equals 1 when it is used. By linearity, the expected number of used days is 365 times that per-day chance: E[	ext{distinct days}] = 365 	imes \left(1 - \left(	frac{364}{365}ight)^{100}ight) \approx 365 …

How many distinct birthdays among a hundred people?

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