The birthday paradox

We’ll ignore February 29 for the purposes of this problem.

There’s a 50% chance that two people share a birthday.
Once a population hits 366 people, it is statistically guaranteed that two people have the same birthday since (for the purposes of out problem) there are only 365 possible birthdays. However, assuming that all birthdays are equally likely, once you have 57 people grouped together, there is a 99% chance that two of them have the same birthday. How do we figure this out?

Let’s look back at our 23-person office to understand how this is possible. We’re going to calculate the converse probability -that no two people in the group share the same birthday- to figure out what we want. Figuring out the probability that at least two people in the office have the same birthday is difficult if you attack it head on, but figuring out the probability that nobody in a group of people has the same birthday is much, much easier.

The probability that two people don’t have the same birthday is this:

The probability that three people don’t have the same birthday is this:

The probability that four people don’t have the same birthday is this:

See where we’re going with this? So, the probability that 23 people don’t have the same birthday is:

This means that since there is a 49,3% chance that nobody has the same birthday, there’s a 50,7% chance that at least two people have the same birthday.

Here’s what the probability curve looks like:

Source: thisinsider.com