The Birthday Paradox

What is the probability that at least two persons have the same birthday (month and day) in a group of 25 people?

First compute the probability of having no pairs within the 25 people. Imagine that the 25 people are ordered from number 1 to number 25. For individual 1, the probability of having a distinct birthday from the previous individual is 1 (there are no previous individual). For the second individual, the probability of having a distinct birthday from the previous individual is 364/365 (there is 1 chance over 365 that she has the same birthday). For the third individual, the probability of having a distinct birthday from the previous 2 individuals is 363/365 and so forth.

The joint probability of seeing no pairs in a group of 25 people is (this is simply the product of all the individual probabilities):

 1 * 364/365 * 363/365 * ... * 1/365 = 43.1%

Then, the probability of seeing at least one pair is 1 - 0.431 = 56.9%. This is more than you would think. Try it on your friends.

This is known as the Birthday Paradox. There are plenty of references to it on the web.

Wikipedia link to the Birthday Paradox