How many people does it take for two to share a birthday?

You need only 23 people. With 23 people in a room, the chance that at least two of them share a birthday is just over 50%, about 50.7%. With 57 people it climbs past 99%. Most of us would guess a number closer to half of 365, which is why this is called the birthday paradox.

The trick is that birthdays aren't compared with yours alone: they are compared between every possible pair of people. You, in a room with 22 others, get only 22 comparisons. But the whole group of 23 contains 253 different pairs, and each one is a fresh chance for a match.

Any single pair is very unlikely to match: two people have a 364 in 365 chance, about 99.7%, of having different birthdays. But multiply that small risk across 253 pairs and the odds that none of them match shrink to about 50%. So a school class of 25 is more likely than not to hide a shared birthday.

And real birthdays, which aren't perfectly evenly spread over the year, don't change the answer: in the real world, 23 is still the magic number.