What Is the Birthday Paradox?

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.

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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.

birthday paradox

This graph shows the probability that there is at least one pair of people with the same birthday among a certain number of people.

Wikimedia Commons (CC BY SA 3.0)

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.

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