BirthdayImagine a room of 23 strangers. How likely is it that two of them share a birthday? Most people guess somewhere around 6 percent, since 23 is a tiny fraction of the 365 days in a year. The real answer is just over 50 percent. This is the birthday paradox, and it is one of the best examples of how badly human intuition handles probability.
Why it is called a paradox
It is not a paradox in the logical sense, only a result that feels wrong. The usual mistake is to think about the chance that someone shares your birthday, which is a different and much less likely event. The birthday paradox asks whether any two people in the group match, and there are far more ways for that to happen.
The pairs that make the difference
In a group of 23 there are 253 different pairs of people (23 times 22, divided by 2). Each pair has a 1 in 365 chance of matching, and with 253 chances the odds add up quickly. The number of pairs grows much faster than the number of people, which is why the probability climbs so steeply as a group gets bigger.
The numbers
The easiest way to calculate it is to work out the chance that nobody shares a birthday and subtract that from 1. With 23 people, the chance of at least one match is about 50.7 percent. With 30 people it is about 70 percent, with 50 people about 97 percent, with 57 people about 99 percent, and with 70 people about 99.9 percent. These figures assume 365 equally likely birthdays and ignore February 29. Real birthdays are not perfectly evenly spread across the year, which makes matches slightly more likely than these figures suggest. A match is only guaranteed with 367 people, because there are 366 possible birthdays once you include February 29.
How to calculate it yourself
Multiply the chance that each new person avoids all the earlier birthdays: 365/365 × 364/365 × 363/365, and so on, one fraction per person. For 23 people you multiply 23 fractions and get about 0.493, which is the chance that all 23 birthdays are different. Subtract that from 1 and you get 0.507, the chance of at least one shared birthday. Any spreadsheet can do this with a column of formulas.
What does not change
The chance that someone shares your specific birthday stays small. In a group of 23 including you, the chance that one of the other 22 people shares your birthday is only about 6 percent, and you need around 253 other people before it reaches 50 percent. The paradox is about any match inside the group, not a match with you.
Try it yourself
Test it in a school class, an office or a sports team: in groups of 30 to 40 people, a shared birthday turns up more often than not. You can also explore your own birth date with our Birthday Calculator, or find out how many days remain until a friend's birthday with our Days Until Birthday calculator, which handles February 29 birthdays correctly.