The Birthday Paradox

Pairs, not people

In a room of 23 people, what is the chance that at least two of them share a birthday? Take every birthday to be equally likely to fall on any of 365 days.

The answer is a little more than a half, which surprises most people. The instinct is to compare one person with the 22 others. But a match can come from any two people in the room, so what matters is the number of pairs.

Each of the 23 people can be paired with each of the other 22, which gives 23 × 22 = 506. That counts every pair twice, once from each end, so there are 506 ÷ 2 = 253 pairs. In the same way, 5 people make 5 × 4 ÷ 2 = 10 pairs.

n people make $n\times \left(n-1\right)/2$n × (n − 1) / 2 pairs. The people go up by a factor of about 11 from 5 to 57, and the pairs go up by a factor of about 160.

A rough estimate

One pair of people share a birthday with probability $\frac{1}{365}$1/365: whatever the first person's birthday is, the second person's is the same day 1 time in 365.

With 5 people there are 10 pairs, so a rough estimate of the chance of a match is $10\times \frac{1}{365}=\frac{10}{365}$10 × 1/365 = 10/365, which is about 0.027, or about 3 in 100. It is rough because the pairs share people: if A matches B and B matches C, then A matches C as well, so the pairs are not separate chances.

Everyone different, one person at a time

The exact answer comes from the complement. At least one match is the opposite of all the birthdays being different, so find the chance that all are different and take it away from 1.

Line the 5 people up. The first can have any birthday: $\frac{365}{365}$365/365. The second must avoid the first person's day: $\frac{364}{365}$364/365. The third must avoid two days, $\frac{363}{365}$363/365, the fourth three days, $\frac{362}{365}$362/365, and the fifth four days, $\frac{361}{365}$361/365. The chance that all five are different is $\frac{365}{365}\times \frac{364}{365}\times \frac{363}{365}\times \frac{362}{365}\times \frac{361}{365}$365/365 × 364/365 × 363/365 × 362/365 × 361/365, which is about 0.973.

So the chance of at least one match is 1 − 0.973 = 0.027, the 3 in 100 that the estimate gave. In 97 rooms of 5 people out of 100, nobody matches.

Out of 100 rooms of 5 people, about 97 have no shared birthday. The 3 squares left over are the rooms with a match.

Twenty-three people

For 23 people the chain has 23 fractions, one for each person, from $\frac{365}{365}$365/365 down to $\frac{343}{365}$343/365. Each fraction is close to 1, but they multiply, and each one is a little smaller than the one before. The product is about 0.493.

So the chance of at least one match is 1 − 0.493 = 0.507. With 23 people a shared birthday is slightly more likely than not.

The same chain gives every room size. With 10 people the chance of a match is about 0.12, with 30 people about 0.71, with 40 people about 0.89, and with 57 people about 0.99. The chance never reaches 1 while there are 365 people or fewer, because they could all still have different birthdays. With 366 people there are more people than days, so a match is certain.

The chance of at least one shared birthday against n, the number of people. The curve crosses the level line at one half between 22 and 23 people, and it is at 0.99 with 57 people.

Someone with your birthday

A different question has a much smaller answer. In a room with you and 22 other people, what is the chance that someone shares your own birthday? Each of the others misses your day with probability $\frac{364}{365}$364/365, so the chance that all 22 miss it is ${\left(\frac{364}{365}\right)}^{22}$(364/365)²², which is about 0.941. The chance that someone shares your birthday is about 1 − 0.941 = 0.059, about 6 in 100.

For that chance to pass one half, you need 253 other people: $1-{\left(\frac{364}{365}\right)}^{253}$1 − (364/365)²⁵³ is about 0.500. That is the same number as the pairs in a room of 23. A room of 23 gives 253 pairs, and each pair is one chance to match.

The usual mistakes

Comparing 23 with 365 and expecting a chance of about $\frac{23}{365}$23/365. That is close to the chance that someone matches one particular person. Any of the 253 pairs can match.

Giving the product of the chain as the answer. $\frac{365}{365}\times \frac{364}{365}\times \dots$365/365 × 364/365 × … is the chance that all the birthdays are different. The chance of at least one match is 1 minus it.

Adding the pair chances as if they were separate: $253\times \frac{1}{365}$253 × 1/365, which is about 0.69 for 23 people. The pairs share people, so the sum counts some matches more than once. With 28 people the sum would be $\frac{378}{365}$378/365, more than 1, which no probability can be.

Days of the week

In the application below, part (a) runs the same chain for three friends and the 7 days of the week, and part (b) finishes the calculation for 23 people.