The birthday paradox
Human intuition is terrible at probability — and the birthday paradox is the most elegant demonstration of it. It's not a trick question or wordplay: it's an exact result that almost everyone misses by a huge margin. Let's compute it and test it.
1. How many people do you think it takes?
A room fills up with people at random. How many people must be in the room for there to be a 50% chance that at least two of them share the same birthday? (Ignore leap years and assume birthdays are evenly distributed.)
2. The curve nobody expects
The probability doesn't creep up — it takes off. Drag the number of people and follow along. At 23 it already passes 50%; at 50 people it's practically certain (97%); at 70, 99.9%.
Curve of shared-birthday probability as the number of people grows.
3. Don't take my word for it — run the experiment
The theory is elegant, but you can test it. The button below simulates real rooms: for each room it draws a random birthday for every person and checks for a match. Run thousands of rooms and compare the empirical result against the formula.
Because you think about yourself. The question feels like "how many people until someone shares my birthday?" — and that answer really is high: 253 people for a 50% chance. But the paradox doesn't ask that: it asks whether any two match. With 23 people there are 253 possible pairs — and it's the number of pairs, not people, that grows fast (it grows with the square). The same illusion explains why "amazing coincidences" happen all the time: there are far more chances to match than we imagine.
Compute other probabilities
Simple and compound probability, with and without replacement — formula explained.
Open the Probability Calculator →See also: Combinations & permutations · Lesson: the average lies