Birthday Paradox Calculator

It is the surprising result that in a group of just 23 people there is a better than even chance that two of them share a birthday. It feels wrong because we picture matching our own birthday, but the paradox counts every possible pair, and 23 people form 253 pairs.

It is easier to find the chance that everyone has a different birthday, then subtract from 1. The first person is free, the second avoids 1 day, the third avoids 2, and so on, giving a product of (365 minus i) over 365. One minus that product is the chance of a shared birthday.

Because the number of pairs grows quickly. With 23 people there are 23 times 22 divided by 2, which is 253 pairs, and each pair has a small but nonzero chance of matching. Added across all those pairs the probability crosses 50 percent right around 23 people.

With 365 possible birthdays, 366 people guarantee a match by the pigeonhole principle, since there are more people than days. In practice the probability is already above 99.9 percent by around 70 people, so a match is nearly certain well before that.

Yes, the standard calculation treats all 365 days as equally likely and ignores leap years and seasonal patterns. Real birthdays cluster slightly, which only makes shared birthdays a little more likely, so the even-spread answer is a close and slightly conservative estimate.