# What Is the Birthday Paradox?

By: Laurie L. Dove  |

How many people do you share a birthday with? For many years, I didn't know anyone who shared my birthday, but as my group of acquaintances expanded, so too did the probability that at least some of them would share the same date of birth. Now I know at least five other people with the same summer birthday as mine. What are the odds?

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## How Does the Birthday Paradox Work?

The answer lies within the birthday paradox: How large does a random group of people have to be for there to be a 50 percent chance that at least two of the people will share a birthday?

Take a classroom of school children, for example. Let's say there are 30 children in the class who have 365 possible birth dates in a calendar year. The odds that any of the students would share a birthday seem pretty low, right? After all, in a group of only 30 children, whose arrivals were randomly spread over 10 times as many days throughout a year, none would probably share a birth date, right?

So, just how large does a group of random people need to be in order for two of them to share a birthday? Most people who quickly do the mental math will believe 182 is the correct answer, which is roughly half the number of days in a year. But would you really need 182 people in a group for two of them to have the same date of birth?

Nope, it's not that simple: The birthday paradox deals in exponentials.

## The Probabilities of the Birthday Paradox Are Exponential

"Most importantly, people 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," Jim Frost, a statistician and columnist for the American Society of Quality's Statistics Digest, told Live Science.

We're just not that great at guessing probabilities, especially when they are as counterintuitive as the birthday paradox.

"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," Frost said.

To understand the probable number of people in order for two of them to be birthday twins, we have to do the math — and begin a process of elimination.

For a group of two people, for example, the chance that one person will share a birthday with the other is 364 out of 365 days. This is a probability of about 0.27 percent. Add a third person to the group, and the chance of sharing a birthday shifts to 363 out of 365 days, which is a probability of about 0.82 percent.

As you may have guessed — and rightly so — the larger the group, the greater the odds that two people were born on the same day. So what is the right answer to the birthday paradox? If we keep doing the math, we'll discover that when we reach a group of 23 people, there will be about a 50 percent chance that two of them will share a birthday.

Why does 23 seem like such a counterintuitive answer? It all has to do with exponents. Our brains don't generally calculate the compounding power of exponents when we do the math in our heads. We tend to think that calculating probabilities is a linear exercise, which couldn't be further from the truth.

In a room with 22 other people, if you compare your birthday with the birthdays of the other 22 people, it would make for only 22 comparisons.

But if you compare all 23 birthdays against each other, it makes for many more than 22 comparisons. How many more? Well, the one person has 22 comparisons to make, but the second person was already compared to the first person, so there are only 21 for that person to make. The third person then has 20 comparisons, the fourth person has 19, and so on. If you add up all possible comparisons, the total is 253 comparisons, or comparison combinations. Thus, an assemblage of 23 people involves 253 comparison combinations, or 253 chances for two birthdays to match.

Here's another exponential growth problem similar to the birthday paradox. "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."

Mathematical probability questions like these "show how beneficial mathematics can be at improving our lives," Frost said. "So, the counterintuitive results of these problems are fun, but they also serve a purpose."

The next time you're part of a group of 23 people, you can feel confident that you have a 50 percent chance of sharing a birthday with someone.