The birthday paradox
In a class of just 23, it's more likely than not that two share a birthday.
Often credited to Richard von Mises, 1939
Class 9C's bet
On the first day of term in Class 9C, Mr Fernandes made a bet. "There are 30 of you," he said. "I bet you a box of laddoos that at least two of you share a birthday."
The class laughed. There are 365 days in the year and only 30 of them. Surely the teacher would lose.
One by one they called out their birthdays. 14 March. 2 July. 30 October… On the nineteenth name, Priya and Sameer both shouted "8 August!"
Mr Fernandes did not have to buy the laddoos. With 30 people, he explained, his chance of winning was about 70 per cent. With just 23, it's already better than evens.
The puzzle
It takes surprisingly few people for a shared birthday to become likely.
See it for yourself
What's really going on
The trick is that you are not comparing one person with everyone else; you are comparing every possible pair. With 23 people there are 23 × 22 ÷ 2 = 253 pairs, and each pair has a small chance of matching. Lots of small chances add up quickly.
It is easier to work out the chance that nobody shares. The second person must avoid 1 birthday (364/365), the third must avoid 2 (363/365), and so on. Multiply those together for 23 people and you get about 0.493, so the chance of at least one match is about 50.7 per cent. With 70 people it is 99.9 per cent. The same maths is used to test computer security, where it's called a birthday attack.
Think about it
How many people would you need for a better-than-even chance that someone shares your own birthday? (It's far more than 23.)