The Birthday Paradox

You walk in, look around, count roughly. Thirty people. Three hundred and sixty-five days in a year. The odds of a shared birthday feel remote — a lottery, almost. You would not bet on it. Almost everyone guesses somewhere around a hundred and eighty people before a match becomes likely. Some go higher. The instinct is not stupid. You are thinking about yourself. You scan the room and ask: what are the odds that someone here shares my birthday? With thirty people, that is about one in twelve. Unlikely. Your gut says the room is safe. Your gut is counting people. The gut is counting the wrong thing. Picture twenty-three people standing in a circle. Now draw a line between every pair who could compare birthdays — every two people who might look at each other and say, wait, when's yours? Start with the first person. They can compare with twenty-two others. The second person has already compared with the first, so they compare with twenty-one more. The third with twenty more. Keep going.