What is the birthday paradox?

Here's a fun brain teaser: How large does a random group of people have to be for there to be a 50% chance that at least two of the people will share a birthday? The answer is 23, which surprises many people. How is this possible?

When pondering this question, known as the "birthday problem" or the "birthday paradox" in statistics, many people intuitively guess 183, since that is half of all possible birthdays, given how there are generally 365 days in a year. Unfortunately, intuition often fares poorly at this kind of statistical problem.

"I love these types of problems because they illustrate how humans are generally not good with probabilities, leading them to make incorrect decisions or draw bad conclusions," Jim Frost, a statistician who has written three books about statistics and is a regular columnist for the American Society of Quality's Statistics Digest, told Live Science in an email. "Additionally, they show how beneficial mathematics can be at improving our lives. So, the counterintuitive results of these problems are fun, but they also serve a purpose."

To calculate the answer to the birthday problem, Frost began with a few assumptions. First, he disregarded leap years, since that simplifies the math and does not change the results by much. He also assumed that all birthdays have an equal chance of happening.

If you start with a group of two people, the chance the first person does not share a birthday with the second is 364/365. As such, the likelihood they share a birthday is 1 minus (364/365), or a probability of about 0.27%.

If you assume a group of three people, the first two people cover two dates. This means the chance the third person does not share a birthday with the other two is 363/365. As such, the likelihood they all share a birthday is 1 minus the product of (364/365) times (363/365), or a probability of about 0.82%.

The more people in a group, the greater the chances that at least a pair of people will share a birthday. With 23 people, there is a 50.73% chance, Frost noted. With 57 people, there is a probability of 99%.

"I have received messages from college statistics professors who will make a \$20 bet about two people sharing a birthday in a particular statistics class," Frost said. "Given the probabilities associated with the birthday problem, he knows that he's practically guaranteed to win. But every semester, the students always take on the bet and lose! Fortunately, he says he returns the money but then teaches them how to solve the birthday problem."

There may be several reasons the answer to the birthday problem feels counterintuitive. One is that people may unconsciously calculate what the chances are that someone else in a group has their birthday, as opposed to the actual question, which is whether anyone in a group shares a birthday, Frost said.

"Second, I think they also start with something along the lines of, well, there are 365 days in a year, so you probably need about 182 people for a 50% chance," Frost said. "But most importantly, they significantly underestimate how quickly the probability increases with group size. The number of possible pairings increases exponentially with group size. And humans are terrible when it comes to comprehending exponential growth."

The birthday problem is conceptually related to another exponential growth problem, Frost noted. "In exchange for some service, suppose you're offered to be paid 1 cent on the first day, 2 cents on the second day, 4 cents on the third, 8 cents, 16 cents, and so on, for 30 days," Frost said. "Is that a good deal? Most people think it's a bad deal, but thanks to exponential growth, you'll have a total of \$10.7 million on the 30th day."

Originally published on Live Science.