Gather 23 people in a room, chosen at random, and there is a better than even chance that two of them share a birthday. Not one of them matching you, but some pair among them matching each other. Most people guess you would need a crowd of a hundred or more. The real number is 23, and it is one of the most reliable ways to win a friendly bet.
The instinct that fights this is easy to name. You think about yourself. In a room of 23, only 22 other people could match your birthday, and 22 out of 365 does feel like long odds. But the puzzle is not asking about you. It is asking whether any two people match, and that is a different question with far more chances hiding inside it.
Count the pairs. With 23 people, the number of possible pairings is not 23 but 253, because every person can be matched against every other. Each of those 253 pairs is its own small chance to collide on a birthday. Line up 253 chances and the odds of at least one match climb quickly, past the halfway mark right at 23 people, to a probability of 50.7 per cent.
The climb keeps going, and it gets steep. Add a few more people and the odds run up into the ninety-per-cent range. By the time you have 70 people in the room, a shared birthday is a near certainty, at 99.9 per cent, even though 70 is still only about a fifth of the days in a year.
Nothing is being cheated here. The calculation is easier to run backwards: work out the chance that everyone has a different birthday, with each new person forced to dodge all the dates already taken, and watch that «all different» probability slide below one half as the twenty-third person walks in. What feels like a paradox is just our habit of counting people when we should be counting pairs.
