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Birthday Paradox Calculator

Find the probability that at least two people in a group share a birthday — the surprising result behind the birthday paradox.

Number of people

Days in the year

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How it works

P(all different) = ∏ (from i=0 to n−1) of (days − i) ÷ days · P(shared) = 1 − P(all different)

The paradox is that a shared birthday becomes likely far sooner than intuition suggests, because it depends on pairs, not individuals — a group of n people has n(n−1)/2 pairs, each a chance to match. It is easiest to compute the opposite: the probability that everyone is different, by multiplying the shrinking fraction of still-free days. Subtract from 1 for the chance of a match. With 365 days it crosses 50% at just 23 people.

Worked example

For 23 people and 365 days, P(all different) = (365/365)(364/365)(363/365)…(343/365) ≈ 0.4927. So P(at least two share) ≈ 1 − 0.4927 = 50.7% — a coin-flip from only 23 people.

Sources: Birthday problem — Wikipedia

Frequently asked questions

Why does it only take 23 people to reach 50%?

Because you are counting pairs, not people. 23 people form 253 different pairs, and each pair is a fresh chance for a match. Our intuition wrongly compares one person against the rest instead of every pair against every other.

How many people guarantee a shared birthday?

By the pigeonhole principle, 366 people guarantee a match with 365 possible birthdays (367 once you include Feb 29). The probability reaches about 99.9% by just 70 people, long before certainty.

Does this assume birthdays are evenly spread?

Yes — it assumes every day is equally likely and ignores leap years and seasonal birth patterns. Real birthdays cluster slightly, which very marginally increases the chance of a match, so 23 is a mild overestimate of what is needed.

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This calculator is for educational purposes. Double-check important results before acting on them.