by Here’s a fun brain teaser: how large does a random group of people need to be so that there is a 50% chance that at least two of the people share a birthday? The answer is 23, which surprises many people. How is that possible?
Known in statistics as the “birthday problem” or “birthday paradox,” many people intuitively guess 183 because that’s half of all possible birthdays, since the year typically has 365 days. Unfortunately, intuition often fares poorly on this type of statistical problem.
“I love these kinds of problems because they illustrate that people are generally not good at dealing with probabilities, which causes them to make wrong decisions or draw bad conclusions.” Jim Frost (opens in new tab), a statistician who has written three books on statistics and is a regular columnist for the American Society of Quality’s Statistics Digest, told Live Science in an email. “In addition, they show how beneficial mathematics can improve our life. So the counter-intuitive results of these problems are fun, but they also serve a purpose.”
To calculate the answer to the birthday problem, Frost started with some assumptions. At first he ignored leap years, as this simplifies the math and doesn’t change the results significantly. He also assumed that all birthdays have an equal chance.
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If you start with a group of two people, the probability that the first person does not share a birthday with the second person is 364/365. Therefore, the probability that they share a birthday is 1 minus (364/365), or about a 0.27% probability.
Assuming a group of three people, the first two people cover two appointments. This means that the probability that the third person does not share a birthday with the other two is 363/365. Therefore, the probability that they all have a birthday is 1 minus the product of (364/365) times (363/365), or about a 0.82% probability.
The more people there are in a group, the greater the chances that at least two people will share a birthday. 23 people have a 50.73% chance, Frost found. With 57 people, the probability is 99%.
“I’ve had messages from college stats professors making a $20 bet on two people who have a birthday in a certain stats class,” Frost said. “Given the probabilities associated with the birthday problem, he knows he’s virtually guaranteed to win. But every semester the students take the bet and lose! Luckily, he says he’ll return the money, but then he teaches them how to solve the birthday problem.”
There can be several reasons why the answer to the birthday problem doesn’t feel intuitive. One is that people subconsciously calculate how likely it is that someone else in a group has a birthday, as opposed to actually asking whether someone in a group has a birthday, Frost said.
“Second, I think they’re also starting out with something like, well, there’s 365 days in the year, so you probably need about 182 people for a 50 percent chance,” Frost said. “But above all, they clearly underestimate how quickly the probability increases with group size. The number of possible pairings increases exponentially with group size. And humans are terrible at understanding exponential growth.”
The birthday problem is conceptually linked to another exponential growth problem, Frost noted. “In exchange for a service, 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, etc. for 30 days,” Frost said. “Is that a good deal? Most people think it’s a bad deal, but thanks to the exponential growth, by day 30 you have a total of $10.7 million.”
Originally published on Live Science.
