Put just 23 people in a room and a shared birthday becomes likely.
Ask people how many strangers you need for a better-than-even chance that two share a birthday, and most guess hundreds or thousands. The real answer is 23. The trick is that you are not comparing birthdays to yours, but every person against every other.
The birthday problem asks a simple question: in a group of randomly chosen people, what is the chance that at least two share a birthday? Assume 365 equally likely days, ignore leap years and twins, and the probability passes 50 per cent at just 23 people, fewer than one fifteenth of the days in a year. Mathematicians call it a veridical paradox: it feels wrong, yet it is true.
The intuition fails because we picture ourselves in the room, hunting for someone who matches us. But the question counts any match between any pair. Twenty-three people form 253 distinct pairs, and every one of them is a fresh chance for a coincidence. The easiest way to calculate it is backwards: work out the odds that everyone's birthday is different, multiplying 364/365 by 363/365 and so on, then subtract from one. For 23 people that 'all different' figure dips to about 49.3 per cent.
The question you were probably imagining has a very different answer. For a better-than-even chance that someone in the room shares your particular birthday, you need at least 253 other people. Real birthdays are not spread perfectly evenly through the year, but the unevenness makes matches slightly more likely, not less, and in practice the magic number stays at 23.
The puzzle is usually credited to the mathematician Harold Davenport around 1927, though he never published it and doubted he was the first to think of it. Richard von Mises put a version in print in 1939. It is more than a party trick: cryptographers use the same arithmetic in the 'birthday attack', which exploits how quickly collisions appear when many items are compared pairwise.
Source: Wikipedia — Birthday problem · Text summarised from Wikipedia (CC BY-SA 4.0)
