The Birthday Paradox

Consider the following example: Assuming that birthdays are evenly distributed throughout the year, if you’re sitting in a room with forty people in it, what are the chances that two of those people have the same birthday? For simplicity’s sake, we’ll ignore leap years. A reasonable, intelligent person might point out that the odds don’t reach 100% until there are 366 people in the room (the number of days in a year + 1)… and forty is about 11% of 366… so such a person might conclude that the odds of two people in forty sharing a birthday are about 11%. In reality, due to Math’s convoluted reasoning, the odds are about 90%. This phenomenon is known as the Birthday Paradox.

If the set of people is increased to sixty, the odds climb to above 99%. This means that with only sixty people in a room, even though there are 365 possible birthdays, it is almost certain that two people have a birthday on the same day. After making these preposterous assertions, Math then goes on to rationalize its claims by recruiting its bastard offspring: numbers and formulas.

It’s tricky to explain the phenomenon in a way that feels intuitive. You can consider the fact that forty people can be paired up in 780 unique ways, and it follows that there would be a good chance that at least one of those pairs would share a birthday. But that doesn’t really satisfy the question for me, it just feels marginally less screwy. So I did something quite out of character: I crunched the numbers. The values rapidly become unmanageable, but the trend is clear: Here is the Whole Story.