Mala Mahadevan explains the Birthday problem and demonstrates it with SQL and R:

Given a room of 23 random people, what are chances that two or more of them have the same birthday?This problem is a little different from the earlier ones, where we actually knew what the probability in each situation was.

What are chances that two people do NOT share the same birthday? Let us exclude leap years for now..chances that two people do not share the same birthday is 364/365, since one person’s birthday is already a given. In a group of 23 people, there are 253 possible pairs (23*22)/2. So the chances of no two people sharing a birthday is 364/365 multiplied 253 times. The chances of two people sharing a birthday, then, per basics of probability, is 1 – this.

The funny thing for me is that I’ve had the Birthday problem explained three separate times using as a demo the 20-30 people in the classroom. In none of those three cases was there a match, so although I understand that it is correct and how it is correct, the 100% failure to replicate led a little nagging voice in the back of my mind to discount it.