The Birthday Paradox: Why 23 People Is Enough

The surprising answer

How many people need to be in a room before there is a 50% chance that at least two of them share a birthday? Most people guess somewhere around 180 — half of 365. The real answer is 23.

It is called a paradox only because it clashes with intuition. The mathematics is straightforward.

Count the pairs, not the people

Our instinct is to think about our own birthday. But the question asks about any two people. With 23 people there are 23 × 22 ÷ 2 = 253 different pairs, and each pair is a separate chance for a match. Two hundred fifty-three small chances add up quickly.

The calculation

It is easier to calculate the chance that nobody shares a birthday, then subtract from 100%. The second person must avoid the first person’s birthday (364 of 365 days), the third must avoid both (363 of 365), and so on:

P(no match among n) = (365 × 364 × … × (365 − n + 1)) ÷ 365ⁿ

For 23 people, that product is about 0.493, so the chance of at least one match is about 1 − 0.493 = 0.507, or 50.7%.

What about your own birthday?

That is a very different question. The chance that at least one of the 22 other people shares your specific birthday is only 1 − (364/365)²² ≈ 5.9%. To reach a 50% chance of someone matching you, you would need about 253 other people.

Assumptions and real-world uses

The classic calculation assumes 365 equally likely birthdays and ignores February 29. Real birthdays are not perfectly uniform — some months see more births — and uneven distributions actually make shared birthdays slightly more likely, not less.

The same ‘many pairs’ logic appears in computer security. A so-called birthday attack exploits the fact that collisions between digital fingerprints (hashes) appear far sooner than intuition suggests, which is why security standards use very long hash values.