Suppose that there are N people in the room Find the formula

Suppose that there are N people in the room. Find the formula for the probablity that 2 or more of them share the same birthday?

Solution

Note that

P(2 or more have the same bday) = 1 - P(all birthdays are unique)

Ignoring leap years, we have P(365, N) ways to have N distinct birthdays.

There are 365^N ways to have all possible birthdays.

Hence,

P(all bdays are unique) = P(365, N) / 365^N or 365! / [(365 - N)! 365^N]

Thus,

P(2 or more have the same birthday) = 1 - {365! / [(365 - N)! 365^N]} [ANSWER]