Definition
The birthday paradox is the surprising fact that in a group of just 23 randomly chosen people, the probability that at least two share a birthday exceeds 50%. With 50 people, the probability rises to about 97%; with 70, it is over 99.9%. The result is paradoxical only in the loose sense — it contradicts intuition, but the mathematics is straightforward.
The puzzle illustrates how rapidly the number of pairs grows with group size: with n people there are n(n-1)/2 pairs, each with a 1/365 chance of matching, and the matching probability grows accordingly.
Why it matters
How it works
The cleanest computation goes through the complement. The probability that all n people have different birthdays is (365/365) × (364/365) × (363/365) × ... × ((365 - n + 1)/365). Subtract from 1 to get the probability of at least one match. For n = 23, the result is about 0.507.
For n people in a uniform population of N possible values, the same formula gives matching probability around 1 - exp(-n²/(2N)) for moderate n — which crosses 50% around n ≈ 1.18 × square root of N. This square-root scaling explains why birthday-type collisions are far more common than naive intuition suggests.