I have 2,000 Facebook friends and noticed that not one of them has a birthday today (January 10th). I quickly calculated the probability of this as (364/365) ^ 2000 = 0.4% (assuming, for the purposes of this calculation, that each birthday is equally likely).
Here's my question, though: how would I calculate the probability of nobody (out of my 2,000 friends) having a birthday on at least one of the 365 calendar days? In other words, what are the odds that I don't have a birthday on each of the 365 days in the year?
I realize it would be 1-P(birthdays on all 365 days); that said, I have no idea how to calculate P(birthdays on all 365 days) either.
Please assume, for the purposes, of this calculation that there are no leap days and that each birthday is equally likely.