On Facebook, I can see which friends have birthdays today. Sometimes there are 1, sometimes more than 1, and sometimes zero friends. What's the average number of birthdays today? To formalize:
I have $n$ friends. Each of their birthdays are equally likely to occur on any day. Let $X$ be the number of friends with birthdays today. Ignore leap years.
Clearly, $0 \leq X \leq n$, with the mean being $n/365$.
What does the probability mass function of $X$ look like? Is this is a Poisson distribution? If yes, what is $\lambda$? If no, how do I describe the distribution?