How many people does it take for two to share a birthday?
How many people does it take?
How many people must be in a room for it to be more likely than not that two share a birthday? A year has 365 days, so most people guess around 180.
It’s not about your birthday
A guess of around 180 is half a year, as if someone had to match your birthday. But all it takes is for any two people in the room to match, even two you don’t know at all. That’s a completely different question.
How many pairs are there?
Every pair of people is one chance of a match. The first of 10 people pairs with the other 9, the second with another 8, and so on: 45 pairs in all. With 30 people there are already 435: pairs pile up far faster than people.
One pair hardly ever matches
The chance that two particular people share a birthday is small: 1 in 365, about three tenths of a percent. But among 30 people there are 435 such pairs, and a single match is enough. What does that add up to?
You can’t just add them up
The pairs’ chances can’t simply be added: with 28 people there are 378 pairs, and you’d get more than a hundred percent. It’s easier to work out the opposite, the chance that nobody matches, and subtract it from a hundred percent. What’s left is the chance that at least two match.
Each new person has to dodge
The second person has to have a birthday on a different day from the first, which leaves 364 days out of 365. The third has to avoid two taken days, the fourth three. The second and the third and the fourth all have to miss, so each one cuts a slice off what’s left.
Half at 23
The slices multiply, and each one cuts away more, because more and more days are taken. At 23 people the chance that everyone misses drops below a half. So in a group of 23 people it’s more likely than not that two share a birthday.
And who shares my birthday?
For someone to have exactly the same birthday as you, you need 253 other people, and even then the chance is only 50%. It’s the same number as the pairs among 23 people: in both cases you need about 253 chances of a match.
The more people, the more certain
In a class of 30 children the chance of a match is about 70%, with 50 people it’s 97%, and with 70 people a match is almost certain. There’s nothing strange about these big numbers: the pairs just pile up fast.
See for yourself
Ask about birthdays in a class or at work where there are at least 25 people, not counting twins. The chance of finding two with the same birthday is more than half. If you try it in several groups, you’ll find a match in roughly every other one.
So how many people does it take?
23 is enough. Nobody has to match your birthday; any pair will do, and among 23 people there are 253 pairs. Each has a small chance, but a single match is enough: the chance that all 253 pairs miss is less than a half.