What is the probability that two people have the same birthday? (The Birthday Paradox)

Let’s say there are 23 students in your class.

**What is the probability that two students in your class have the same birthday****?**(we’ll ignore February 29 for the purposes of the problem)

**ANSWER**

**There is a ****50%**** chance …****!!! **

**Are you agree with me****??????**

Once a population hits 366 people, it is statistically guaranteed that two people have the same birthday by **the**^{††}**pigeonhole principle**. 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 the class to understand how this is possible.
- We are 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 students in the class have the same birthday is difficult if you attack it head on.
- Figuring out the probability that nobody in a group of people has the same birthday is much, much easier.

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

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

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

See where we are going with this?

So, the probability that **23** students don’t have the same birthday is:

This means that since there is a 49.3% chance that nobody has the same birthday, there is a 50.7% chance that **at least** two students have the same birthday.

Following graph show the computed probability of at least two people sharing a birthday amongst a certain number of people.

^{††}**pigeonhole principle: **if *n* items are put into *m* pigeonholes with *n*>*m*, then at least one pigeonhole must contain more than one item.

Aruna Bandaranayake