You Say It’s Your Birthday

It’s Terra Cognition’s first actual post – and today, the subject is a math problem that isn’t what it seems.

Imagine that you are at a party with a large group of people. After introductions, the topic turns to ages, specifically birthdays. How many people do you think there would need to be in this group before there is more than a 50% chance that two people shared the same birthday (day and month, not year)?

Intuitively, you might think the number should be rather large. There are 365 possible days to choose from (ignoring February 29 in this example), so there should be a large number of people needed to guarantee a better than 50% chance of a match. Seems reasonable, right?

This question is what is known in mathematics as the birthday problem, sometimes known as the birthday paradox, so named because the answer seems paradoxical to our intuition: you only need 23 people to have more than a 50% chance of a birthday match.

The calculations can be approached several ways, but let’s look at the chance of someone having a different birthday than yours. The chance that someone has a different birthday than yours is inversely related to the chance that your birthdays match; the more people whose birthdays are unique, the less birthdays there are and correspondingly more chances of a match. The chance that someone has a different birthday than you is 364/365 (364 possible different dates / 365 days in a year). The chance that a second person has a different birthday from you and the first person is 363/365.

By multiplying these probabilities together (364/365 x 363/365 x 362/365…), the chance of unique birthdays decreases. By the time you reach the 23rd person, the chance that there are unique birthdays in the group has dropped to 49.3%, meaning the chance of a match is now 51.7%. If you were to gather a groups of 23 people together, more than 50% of the time, there would be a matched birthday.

The probability that there will be two or more birthday matches among groups of people (via Wikipedia)

This calculation assumes that all birthdays are equally probable. They’re not, since birthdays are more likely in certain seasons than others and February 29 occurs only once every four years.

Try it yourself! The next time you meet people at a party with the same birthdays, it may be more than a coincidence: it just might be statistically probable.