How many people does it take for there to be a 50% chance that a pair in the group has the same birthday? Only 23 people. What about a 99% chance? Maybe even more shocking: 57 people. This is the birthday problem, which every undergrad who’s taken a stat course has seen. Steven Strogataz explains the logic and calculations.

Intuitively, how can 23 people be enough? It’s because of all the combinations they create, all the opportunities for luck to strike. With 23 people, there are 253 possible pairs of people (see the notes for why), and that turns out to be enough to push the odds of a match above 50 percent.

Incidentally, if you go up to 43 people — the number of individuals who have served as United States president so far — the odds of a match increase to 92 percent. And indeed two of the presidents do have the same birthday: James Polk and Warren Harding were both born on Nov. 2.

The Johnny Carson clip referenced in the article is worth watching. Carson tries to test the results with the audience, but goes about it the wrong way.

For a different application of the same mathematics, consider the question, “What is the probability that people in the same meeting room share initials?” This question was asked by a colleague who had been taking notes at a meeting. He recorded each person’s initials next to their comments and, upon editing the notes, was shocked to discover that no two people in the meeting shared initials. My analysis (which uses the actual distribution of initials at my company) is at http://blogs.sas.com/content/iml/2011/01/19/hey-those-two-people-have-the-same-initials/

The flip side of this question: what are the odds of having 1800 friends on Facebook and not sharing a birthday with any of them?

0.7 %… Rare but hardly impossible.