# The Birthday Problem is a famous statistical problem that tells us that there is about a 50% chance that, out of just 23 people in a room, at least two of them will share the same birthday (month and day). How is the multiplication rule used to calculate this probability?

##### 1 Answer

Feb 18, 2015

Actually, it is *the subtraction rule* that is more important here.

Let's try to work out the probability that they all have a **different** birthday:

First person may choose 365 days out of 365.

Second person has only 364 choices left,

Etc, etc.

So the chance that they all have different birthdays is:

Which works out to be

So the chance that there is at least one double is:

(I won quite a few bets on that, because it is really counter-intuitive)