Twenty-three People Is Enough for a Shared Birthday

Put twenty-three people in a room. There is a 50.7% chance two of them share a birthday. Not a similar one. The same one.

This sounds wrong because we instinctively think about ourselves. The chance that someone else shares your birthday in a room of twenty-three is genuinely small, about six per cent. But the question is not about you. It is about any pair. Twenty-three people make 253 possible pairs, and each pair has its own small chance of matching. Add those chances up across all 253 pairs and you cross fifty per cent.

The same maths breaks fifty per cent for a 22-person room, and only just. By the time you have seventy people in a room, a shared birthday is 99.9% certain. Most school classes already have one. Most football teams definitely do.

It is called the birthday paradox, even though it is not a paradox. It is a place where our intuition stops being useful. The same thing happens whenever the scale gets larger than our brains were built for. Counting to a billion sounds similar to counting to a million until you do the maths and find out one takes weeks and the other takes a lifetime.

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