Birthday Paradox Calculator

About this calculator

The birthday paradox asks how likely it is that at least two people in a group share a birthday, and the answer is famously counterintuitive: with just 23 people, the probability already exceeds 50%, and with 70 people it’s over 99.9%, even though there are 365 possible birthdays. The exact formula computes the probability of no shared birthday first — 365 ÷ 365 × 364 ÷ 365 × 363 ÷ 365, and so on for each additional person — then subtracts that from 1, since it’s the number of pairs of people, which grows much faster than the group size, that drives the effect.

The same birthday-problem math underlies the birthday attack in cryptography, where finding any collision in a hash function turns out to require far fewer attempts than finding a specific match, which is why hash output lengths are chosen with this square-root-scaling effect in mind. It’s also a standard teaching example in probability courses for demonstrating how intuition about random matching can badly mislead you.

This calculator takes a group size and returns the probability that at least two people in it share a birthday.