The Birthday Problem

# The Birthday Problem: A short lesson in probability.

Last revised 6/2/98

## Introduction with simulation

Happy Birthday! There's a birthday in your class today! Or will there be two? How likely is it that two people in your class have the same birthday? Say your class has 28 students.

There are a number of ways to approach this problem. The most common is to take a survey and see if it happens that two birthdays fall on the same day. But if it happens in the surveyed class, will it occur in another class with different students? The question of how likely it is for any given class is still unanswered.

Another way, is to survey more and more classes to get an idea of how often the match would occur. This can be time consuming and may require a lot of work. But a computer can help out. Below is a simulation of the birthday problem. It takes a little while (about 25 seconds) to respond, but it will generate a random list of birthdays time after time.

Choose a number for your class size and do 10 trials with that size.

Applet Source, Written by Nicholas Exner.

Now what do you think the probability of a match is?

It may surprise you that there were so many matches. Let's look at an explanation for this problem.