Intuitively, you know that the quick mental math example above is wrong. Intuitively, you know the odds are higher than ten percent that two people out of thirty-six have the same birthday. But most of us fail to recognize just how much higher the actual odds are.

To actually calculate probability in the birthday problem, you don’t simply divide the number of all possible birthdays by the number of people in the room.

The actual math too tedious to describe here. What’s important to understand is that each person’s birthday interacts with each other person’s birthday.

If you’re person one, the chances that person two has the same birthday as you are not high: about 1/365—roughly a 2.7% chance. But we’re looking for the chances that any two people

At ten people, the odds have jumped to 11.7%—even more than our rough-math calculation for thirty-six people. At only twenty people: the odds are a whopping 41.1%!

In our example of only thirty-six people, it turns out that it’s almost certain that two of them have the same birthday. The exact odds are an 83.2% chance that two people out of thirty-six have the same birthday.