How big a team before two share a birthday?

Question

Assume nobody was born on February 29, and that birthdays are spread evenly over 365 days.

What is the smallest number of people in a room for which it is more likely than not that at least two of them share a birthday?

Accepted Answer

It is easier to track the chance that everyone's birthday is different, then subtract from 1. Line people up one at a time. The second person avoids the first's birthday with probability 	frac{364}{365}, the third avoids both with 	frac{363}{365}, and so on. For n people, P(	ext{all different}) = \frac{365}{365}\cdot\frac{364}{365}\cdots\frac{365-n+1}{365}. Evaluating: at n = 22 this is about 0.524, so a shared birthday has probability about 0.476, still under a half. At n = 23 the product drops to about 0.493, so P(	ext{shared birthday}) \approx 1 - 0.493 = 0.507 	frac12. So 23 people a…

How big a team before two share a birthday?

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