# 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?

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:

$\frac{365}{365} \cdot \frac{364}{365} \cdot \frac{363}{365} \cdot \ldots . \cdot \frac{343}{365}$

Which works out to be ~~0.49or 49%

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

1-0.49=0.51=51%

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