Birthday Paradox Calculator

Find the probability that at least two people in a group share a birthday. Enter the group size to see the exact chance of a match — it hits 50% at just 23 people.

Quick Facts

Method
P = 1 − (365 × 364 × … × (365−n+1)) / 365ⁿ
Exact probability of at least one shared birthday, assuming 365 equally likely, independent birthdays.

Your Results

Calculated
Chance of a shared birthday
-
At least two people match
Chance everyone is unique
-
No two birthdays coincide
Distinct pairs
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n(n−1)/2 comparisons

Ready

Enter a group size and calculate the probability.

What the birthday paradox is

The birthday paradox (more precisely, the birthday problem) asks: in a group of n people, what is the probability that at least two of them share the same birthday? It is called a paradox because the answer defies intuition — you only need 23 people for the chance to pass 50%, and just 57 people for it to reach 99%. There is nothing self-contradictory about it; the surprise comes from confusing "someone shares my birthday" with "some pair among everyone shares a birthday."

The formula

It is far easier to compute the probability that everyone has a different birthday and subtract from 1. Assuming 365 equally likely, independent birthdays (ignoring February 29 and any seasonal clustering), the chance that all n people are unique is a product of shrinking fractions:

P(no shared birthday) = (365/365) × (364/365) × (363/365) × … × ((365 − n + 1)/365)

P(at least one shared birthday) = 1 − P(no shared birthday)

Equivalently, this is P = 1 − 365! / ((365 − n)! · 365ⁿ). Each new person must dodge every birthday already taken, so the "unique" probability multiplies by one more term that is slightly less than 1. Once n exceeds 365 the probability is exactly 1 by the pigeonhole principle — there simply are not enough distinct days to go around.

Why the number is so small

The key is that matches happen between pairs, not individuals. A group of n people contains n(n−1)/2 distinct pairs. With 23 people that is 253 pairs, each with roughly a 1-in-365 chance of matching — enough opportunities that a hit becomes more likely than not. The count of pairs grows with the square of the group size, which is why the probability climbs so steeply.

Common reference points

  • 10 people → about 11.7% chance of a shared birthday
  • 23 people → about 50.7% — the classic "just past even odds" threshold
  • 30 people → about 70.6% (a typical classroom)
  • 50 people → about 97.0%
  • 57 people → about 99.0%
  • 70 people → about 99.9%

Frequently Asked Questions

Why does it only take 23 people to reach 50%?
Because what matters is the number of pairs, not people. Twenty-three people form 23×22/2 = 253 distinct pairs, and each pair has about a 1-in-365 chance of matching. With 253 chances, the probability that at least one pair coincides climbs just past 50% (about 50.7%). People underestimate this because they picture matches against their own birthday rather than every possible pairing.
Isn't this only about someone sharing MY birthday?
No — that is a different, much rarer question. The probability that someone in a group of n shares your specific birthday is 1 − (364/365)ⁿ⁻¹, which needs about 253 people to reach 50%. The birthday paradox instead asks whether ANY two people match, which is why 23 is enough.
Does the calculator account for leap years or uneven birth rates?
No. The standard formula assumes 365 equally likely, independent birthdays. Real birth dates are slightly clustered (more common in late summer, fewer on Feb 29), which very marginally increases the chance of a match — so the true probability is a hair higher than the model, never lower. For everyday use the difference is negligible. You can enter a different number in the "Days in the year" field (e.g. 366) to explore alternatives.
What happens if the group is larger than 365 people?
The probability is exactly 100%. By the pigeonhole principle, once you have more people than there are days in the year, at least two must share a birthday — there is no way to give everyone a unique day.

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