In a room of 23 people, what is the probability that at least two of them share a birthday? Assume 365 equally likely birthdays and ignore leap years. Give the answer to the nearest percent.

About 51\% — slightly better than a coin flip.

Twenty-Three People in a Room

Issue 32 Warm-up Published 26 Oct 2026 By The Chalkboard Desk

The problem

In a room of people, what is the probability that at least two of them share a birthday? Assume equally likely birthdays and ignore leap years. Give the answer to the nearest percent.

Birthdays are independent and uniform over days.
February 29 is ignored, so every year has days.

Take a minute

Reveal hint 1

Computing 'at least one shared birthday' directly is messy. Compute the complement: the probability that all birthdays are different.

Reveal hint 2

For all birthdays to differ, the second person must avoid date, the third must avoid , and so on.

Solution

AnswerAbout — slightly better than a coin flip.

Derivation

Switch to the complement.
Build the product.
Evaluate numerically.
Subtract from one.
Find the threshold.
Explain the surprise.