Birthday Paradox Calculator

Written byOleh Soloninupdated 5 October 2026

People in the group

Days in the year

Result

Chance of a match50.73%
Chance all differ49.27%
Pairs in the group253
People for 50%23
People for 99%57
What matters
The match is between any two people, not with your own birthday. The number of pairs grows with the square of the group, which is why the probability climbs so fast.

Twenty-three people, even odds

Calculates the chance that two people in a group share a birthday, and how many people are needed for a 50 or 99 percent chance.

The result feels wrong: in a group of 23 there is already a better than even chance that two of them share a birthday. The intuition that goes wrong is comparing everyone with yourself.

The question is about any pair. Twenty-three people make 253 pairs, and it only takes one of them to match. That is why the probability climbs so much faster than the group size.

How to use

Enter the number of people in the group.
Leave 365 days in the year, or set your own figure if you are not counting birthdays.
Read both lines: the chance of a match and the chance that all are different.
Compare against the number of pairs — that is what grows quadratically and explains the whole effect.

Why twenty-three people are enough

Each row is a run of this calculator at 365 days. The chance of a match is worked out the other way round: first the chance that every birthday differs, then one minus that.

The answer is in the last column. What matters is not the number of people but the number of pairs, and that grows as a square: 23 people make 253 pairs, 50 make 1,225. Every pair is its own chance to coincide, which is why the halfway mark arrives so early.

Runs of this calculator at 365 days in the year.
People	Chance of a match	All different	Pairs
10	11.69%	88.31%	45
23	50.73%	49.27%	253
30	70.63%	29.37%	435
50	97.04%	2.96%	1,225
70	99.92%	0.08%	2,415

What this calculator shows

The chance that at least two people share a date
The number of pairs in the group
How many people are needed for a 50 and a 99 percent chance

What to keep in mind

Birthdays are treated as spread evenly across the year, and 29 February is ignored.
Real birthdays cluster slightly by season, which makes matches marginally more likely than the model says.

FAQs

Why 23 and not 183?

Because 183 is just 365 halved, and it answers neither question. Any pair matching is an easy condition: 23 people give 253 pairs. For a 50% chance that someone shares your own birthday, you need 253 other people.

What is the chance someone shares my birthday?

About 6 percent in a group of 23 including you, and it takes 253 other people — 254 with you — to pass even odds. That is the intuition most people are actually using.

Does 29 February change anything?

Almost nothing. Adding a day that occurs a quarter as often shifts the answer by a fraction of a percent.

Where else does this come up?

In cryptography. The same maths says a hash of n bits starts producing collisions after about 2^(n/2) values, which is why 64-bit hashes are not used for security.

Why does twenty-three feel too few?

Because intuition counts matches against your own birthday, while the question asks about any pair at all. You are in 22 pairs; the group holds 253 of them, and that order-of-magnitude gap is where the surprise comes from.

Where else does the same effect turn up?

Anywhere a match is sought inside a set rather than against a fixed value: hash collisions, repeated random identifiers, duplicate reference numbers. The number of pairs to check always grows as the square of the set.

Worked example

A group of 23

Input: 23 people, 365 days

Output: 50.7 percent

Note: At 57 people it passes 99 percent. The chance that someone shares your birthday specifically is only about 6 percent in a group of 23 — a completely different question.

Useful links

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Birthday Paradox Calculator

Result

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