Birthday paradox calculator

There are 23 pupils in a class. What is the chance that at least two of them share a birthday? Most people guess a few per cent, but in fact it is more than half. This is called the birthday paradox. It is not really a paradox, just a case where intuition fails: we are not comparing one person with the others, but everyone with everyone, and there are 253 pairs in the class.

The calculator assumes a year of 365 days and that people are equally likely to be born on any day.

How it works

The calculator uses the formula

$$P=1-\frac{365\cdot 364\cdot \dots \cdot (365-n+1)}{{365}^n}$$

where n is the number of people.

Working this out directly would be messy, since two people, three people or several different pairs could match. So we work with the complementary event instead: the probability that everyone has a different birthday. The first person can have a birthday on any day, the second must hit one of the 364 remaining days, the third one of the 363 remaining days, and so on, up to the 23rd person, who has 343 days to choose from:

$$\frac{365}{365}\cdot \frac{364}{365}\cdot \frac{363}{365}\cdot \dots \cdot \frac{343}{365}\approx 0.4927$$

Subtract from one:

$$P=1-0.4927=0.5073$$

Among 23 people, the probability that at least two share a birthday is about 50.73%. A different question is whether anyone shares a birthday with you. Each of the other 22 people has a different birthday from yours with probability $\frac{364}{365}$, so the chance is only $1-{\left(\frac{364}{365}\right)}^{22}\approx 5.86\mathrm{\%}$.