🎂 Birthday Paradox Calculator
The famous "birthday paradox": in a group of just 23 random people, there's already a better- than-even chance two of them share a birthday. Enter a group size to see the exact probability.
It counts the misses, not the matches
Counting the ways a group could contain a match means counting overlapping pairs, which gets messy quickly. The calculator does the easy half instead: it works out the chance that nobody matches, then subtracts that from 1.
P(no match) = (365/365) × (364/365) × ... × ((365 - n + 1)/365)
P(shared) = 1 - P(no match)Each new person has to dodge every birthday already taken, so the top of each fraction falls by one. With three people the product is 0.9918, giving a 0.82% chance of a match. With 23 people it has fallen to 0.4927, so the answer is 50.73% - the classic result the tool opens on.
Why the odds climb so fast
The comparisons, not the people, drive the number: a group of n makes n(n-1)/2 pairs, so 23 people are being checked against each other 253 times.
| People | Chance of a shared birthday |
|---|---|
| 10 | 11.69% |
| 23 | 50.73% |
| 40 | 89.12% |
| 57 | 99.01% |
| 70 | 99.92% |
Frequently asked questions
How many people do you need for a 50% chance of a shared birthday?
23. At 22 people the chance is 47.57%, and the 23rd person tips it to 50.73%.
Why does 23 feel far too low?
Because intuition answers a different question - how many people it takes for someone to share your birthday, which needs 253 others for a coin-flip chance. The paradox allows any pair to match, and 23 people already form 253 pairs.
What group size makes a shared birthday certain?
366, by the pigeonhole principle: there are only 365 possible birthdays, so the 366th person must repeat one. This tool stops at 365, where the chance already rounds to 100.00%.