### What exactly is the birthday paradox? 16/09/2022

Here’s a fun mental exercise: How large must a random group of people be to have a 50% chance that at least two of the people share a birthday? Many people are surprised by the answer, which is 23. What makes this possible?

When considering this question, known in statistics as the “birthday problem” or the “birthday paradox,” many people intuitively guess 183, because that is half of all possible birthdays, given that there are 365 days in a year. Unfortunately, intuition frequently fails at this type of statistical problem.

“I love these types of problems because they demonstrate how humans are generally bad 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. “They also demonstrate how useful mathematics can be in improving our lives. So, while the counterintuitive outcomes of these problems are entertaining, they also serve a purpose.”

Frost began by making a few assumptions in order to calculate the answer to the birthday problem. First, he ignored leap years because they simplify the math and have little effect on the results. He also assumed that every birthday had an equal chance of occurring.

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

In a group of three, the first two people cover two dates each. This means that the third person has a 363/365 chance of not sharing a birthday with the other two. As a result, the probability that they all share a birthday is 1 minus the product of (364/365) times (363/365), or about 0.82%.

The more people in a group, the more likely it is that at least two people will share a birthday.

Frost noted that there is a 50.73% chance with 23 people. There is a 99% chance with 57 people.

“I’ve received messages from college statistics professors willing to place a $20 bet on two people sharing a birthday in a specific statistics class,” Frost said. “Given the probabilities associated with the birthday problem, he knows he’s almost certain to win. But every semester, the students place the wager and lose! Fortunately, he claims to return the money before instructing them on how to solve the birthday problem.”

There could be several reasons why the solution to the birthday problem appears counterintuitive. One is that people may unconsciously calculate the chances that someone else in a group has their birthday, rather than the actual question of whether anyone in a group shares a birthday, according to Frost.

“Second, I believe they begin 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 explained. “Most importantly, they overestimate how quickly the probability increases with group size. With increasing group size, the number of possible pairings grows exponentially. And humans are terrible at comprehending exponential growth.”

Frost pointed out that the birthday problem is conceptually related to another exponential growth problem. “Suppose you’re offered 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 in exchange for some service,” Frost explained. “Is that a good price? Most people think it’s a bad deal, but thanks to exponential growth, you’ll have $10.7 million by the 30th day.”

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