What is the birthday paradox, and why does 23 people work?

The birthday paradox, also called the birthday problem, asks how likely it is that at least two people in a random group share a birthday. The surprising result is that only 23 people are needed for that probability to pass 50 percent. It is called a veridical paradox: it seems wrong at first glance but is in fact true. The number 23 is less than one fifteenth of the number of days in a year. [1]

Many people guess 183, the smallest whole number above half of 365. According to the NPR column by Keith Devlin, that figure answers a different question: how many people are needed for a better-than-even chance that someone shares your own birthday. When no particular pair is specified, the situation changes enormously. [2]

The difference comes from counting pairs. One person in a room of 23 makes only 22 comparisons, but every person is compared with every other. That gives 22 + 21 + 20 and so on down to 1, which totals 253 pairs, each a chance for a match. [3] [4]

The probability course at probabilitycourse.com makes the same point numerically. For a group of 23, the chance that someone matches your specific birthday is only about 0.0586, while the chance that some pair matches is about 0.5073. [5]

It is easier to calculate the opposite event: that nobody shares a birthday. Two people have different birthdays with probability 364/365. A third person avoids both with probability 363/365, and so on, with each new fraction multiplied in. When the twenty-third person is added, the chance that all birthdays differ drops to about 0.493, so the chance of at least one match is about 0.507. [2]

The same idea can be written as one formula. With 365 equally likely days and k people, the chance of a match is 1 minus the number of no-match arrangements divided by 365 to the power k. The probability passes 99 percent at 57 people. [5]

An alternative route treats the 253 pairs as separate comparisons, each with a 364/365 chance of not matching. Raising that to the 253rd power gives about 49.95 percent for no match, so a match has about 50.05 percent. This is an approximation, since pairs are not truly independent. [3]

The standard version ignores leap years, twins, selection bias and seasonal or weekly variation in births. It assumes 365 equally likely birthdays, independent between people. A uniform spread of birthdays minimizes the chance of a match, but real birthdays are not uneven enough to change much: the group size needed for better than 50 percent is still 23. [1]

One explainer attributes the surprise to our difficulty with compounding, since we tend to expect probabilities to grow in a straight line. [6]