Question

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?

Answer 1

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:

`365/365 * 364/365 * 363/365 * .... *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)