The Birthday Problem

The well-known birthday problem:

How many people do you need to assemble before the probability that some two of them have the same birthday is greater than 0.5?

Assumptions: (1) A year has 365 days (no leap year), and (2) By “same birthday”, we only mean the same day and month (not necessarily the same year).

My first intuition was 183 — the smallest number that is more than half the number of days in a year. But as it turns out, 183 is the number of people you need to assemble before the probability that someone has the same birthday as you — a specific person — is greater than 0.5!

Look closely. The question is not about someone matching a particular person’s birthday. It is about the birthdays of some two people matching. Enough said. What is the right answer then, and why? How do we generalize this to matching the birthdays of some N people?

If you are guessing or have worked out a solution now, please leave comments.

The point of asking such questions (even though they might seem trivial to some) is to further our own understanding, and more importantly, to enjoy the process of understanding! So, if you already know the answer, please defer commenting.