Science Puzzle
The Birthday Surprise
There are 23 people in a room: the players and referee at a football match, say, or a small class with its teacher. Ignore leap years and twins; assume birthdays are spread evenly across the 365 days of the year.
What's the chance that at least two of the 23 people share a birthday?
The Answer
About 50.7%. With just 23 people, a shared birthday is more likely than not.
The surprise comes from counting the wrong thing. Our instinct imagines one person checking their own birthday against everyone else's, which really would give a small chance. But a match can happen between any two people in the room, and the number of possible pairs grows fast. With 23 people there are 23 × 22 ÷ 2 = 253 different pairs, and every one of them is a chance for a match.
The neatest way to do the sum is to work out the chance that nobody matches and take that away from 1. The second person has 364 days out of 365 that avoid the first; the third has 363 out of 365 that avoid both; and so on down to the twenty-third, with 343 out of 365. Multiply all those fractions together and you get about 0.493. So the chance of at least one match is about 1 minus 0.493, which is 0.507: just over a half.
The chance climbs quickly after that: about 70% for 30 people and 99.9% for 70. Your own chance of sharing your birthday with someone in a room of 23, though, really is small, about 6%. It's the difference between "someone matches someone" and "someone matches me".
The principle: The birthday problem. The number of possible pairs grows much faster than the number of people, so coincidences between any two members of a group are far more likely than intuition suggests.