Expected number of different birthdays

(one)Birthday problem!

Jack's Personal Blog on Mathematics

Suppose there are n people in a room. It is the well known ‘birthday problem‘ that if n > 22 the probability of two people sharing the same birthday is greater than 50%, and so it is very likely that there aren’t n unique birthdays. However, what is the expected number of unique birthdays?

[Let us assume there are D days in a year, in order to be more general.]

Actually, the expected number of unique birthdays is very easy to calculate. Let e(n) denote the expected number of unique birthdays when there are n people.

Choose one of the n people. Then either they have a different birthday than everybody else, or they don’t. They have a different birthday than the other n-1 with probability $latex left(frac{D-1}{D}right)^{n-1}&bg=ffffff$. Let us write $latex phi&bg=ffffff$ for $latex (frac{D-1}{D})&bg=ffffff$, and so the probability that they have a different birthday to everybody…

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