Birthday Paradox Calculator
Find the surprising probability that two people in a group share a birthday. Enter the number of people and this calculator returns the chance of at least one shared birthday, the chance of none, the group size needed to cross 50%, and the number of possible pairs.
Formula
P(shared) = 1 − (365/365 × 364/365 × … × (365−n+1)/365)
- It is easier to first find the chance that everyone has a different birthday, then subtract from 1.
- For n people, the chance of all-different birthdays is 365/365 × 364/365 × … × (365−n+1)/365.
- The probability of at least one shared birthday is 1 minus that product.
- It feels paradoxical because there are n(n−1)/2 possible pairs, and that pair count grows fast — with 23 people there are already 253 pairs.
- The model assumes 365 equally likely birthdays and ignores leap years and seasonal birth patterns.
23 people in a room
Inputs
- Number of People: 23
With 23 people there are 253 possible pairs, giving about a 50.7% chance that at least two share a birthday — the famous better-than-even result.
Frequently asked questions
What is the birthday paradox?
It is the surprising fact that in a group of just 23 people there is a better-than-even chance that two of them share a birthday — far fewer people than most expect.
Why does so small a group work?
You are not matching one specific birthday; you are checking every possible pair. A group of 23 has 253 pairs, and each pair is a chance for a match, so the odds add up quickly.
How many people give a 99% chance?
About 57 people push the probability of a shared birthday above 99%, and 70 people make it roughly 99.9%.
What assumptions does it make?
It assumes 365 equally likely birthdays and treats people's birthdays as independent. Real birth dates are slightly uneven and include February 29, which barely changes the result.
What is the crossover point?
It is the smallest group size where the chance of a shared birthday first reaches 50% — which happens at 23 people.