Birthday problem

Suppose we have 23 people in a room. What is the probability that two or more people have the same birthday?

Note

I’m pretty sure this problem is usually phrased the other way around; that is, what’s the minimum number of people needed for the chance that two people share the same birthday to be greater than 50%?

But, Winston introduced it this way, so eh.

Assume all 365 birthdays are equally likely and independent from one another. Also, leap years don’t exist.

We are essentially sampling from 365 choices, 23 times. The size of the sample space is then $|S|=365^{23}$.

Let event $A$ be the event that two or more people share a birthday. This is difficult to count. Instead, calculate $P(A^{\complement })$, the probability that everyone has a different birthday.

Let’s count $A^{\complement }$. If everyone has a different birthday, then we’re simply sampling without replacement. so

$$|A^{\complement }|=\frac{365!}{(365-23)!}$$

From this, we can get $P(A)$.

$$P(A)=1-P(A^{\complement })=1-\frac{(365)(364)(363)...(343)}{365^{23}}\approx 1-0.493=0.507$$

If you have 23 people in a room, the probability that two people share a birthday is greater than 50%. This feels counterintuitive because it feels like a very small number to achieve such a high probability.