The Birthday Paradox Explained
The fascinating math behind shared birthdays.
If you haven’t heard of the birthday paradox, here’s a brief description:
The birthday paradox refers to the bizarre likelihood that a small group of people has at least two people who share the same birthday. For example, in a group of 23 people, there is a 50% chance that two people share the same birthday which is counterintuitive since there are 365 days in the year!
In this article, we’re going to find a formula that takes the size of a group of people and tells us how likely a shared birthday is amongst this group. From here, we’ll explore the likelihood of a shared birthday in groups of all sizes. Let’s begin.
How do we start? Let’s save the general formula for a bit and work through an example — a group of thirty people. Let’s consider one person’s birthday. Now we consider another person, there is a 1/365 chance they share a birthday with the first person. Now we consider another person? We’ll quickly realise that things are going to get a lot harder if we go down this path, since for every new person, we need to compare their birthday to everybody else’s. The best way to calculate the probability of a shared birthday is with the old reliable method:
