Check your intuition: The birthday problem

Imagine a group of people. How big do you think the group would have to be before there’s more than a 50% chance that two people in the group have the same birthday? Assume for the sake of argument that there are no twins, that every birthday is equally likely, and ignore leap years. Take a moment to think about it. The answer may seem surprisingly low. In a group of 23 people, there’s a 50.73% chance that two people will share the same birthday. But with 365 days in a year, how’s it possible that you need such a small group to get even odds of a shared birthday? Why is our intuition so wrong?

To figure out the answer, let’s look at one way a mathematician might calculate the odds of a birthday match. We can use a field of mathematics known as combinatorics, which deals with the likelihoods of different combinations. The first step is to flip the problem. Trying to calculate the odds of a match directly is challenging because there are many ways you could get a birthday match in a group. Instead, it’s easier to calculate the odds that everyone’s birthday is different. How does that help? Either there’s a birthday match in the group, or there isn’t, so the odds of a match and the odds of no match must add up to 100%.

That means we can find the probability of a match by subtracting the probability of no match from 100. To calculate the odds of no match, start small. Calculate the odds that just one pair of people have different birthdays. One day of the year will be Person A’s birthday, which leaves only 364 possible birthdays for Person B. The probability of different birthdays for A and B, or any pair of people, is 364 out of 365, about 0.997, or 99.7%, pretty high. Bring in Person C.

The probability that she has a unique birthday in this small group is 363 out of 365 because there are two birthdates already accounted for by A and B. D’s odds will be 362 out of 365, and so on, all the way down to W’s odds of 343 out of 365. Multiply all of those terms together, and you’ll get the probability that no one shares a birthday. This works out to 0.4927, so there’s a 49.27% chance that no one in the group of 23 people shares a birthday. When we subtract that from 100, we get a 50.73% chance of at least one birthday match, better than even odds. The key to such a high probability of a match in a relatively small group is the surprisingly large number of possible pairs.

As a group grows, the number of possible combinations gets bigger much faster. A group of five people has ten possible pairs. Each of the five people can be paired with any of the other four. Half of those combinations are redundant because pairing Person A with Person B is the same as pairing B with A, so we divide by two. By the same reasoning, a group of ten people has 45 pairs, and a group of 23 has 253. The number of pairs grows quadratically, meaning it’s proportional to the square of the number of people in the group.

Unfortunately, our brains are notoriously bad at intuitively grasping non-linear functions. So it seems improbable at first that 23 people could produce 253 possible pairs. Once our brains accept that, the birthday problem makes more sense. Every one of those 253 pairs is a chance for a birthday match.

For the same reason, in a group of 70 people, there are 2,415 possible pairs, and the probability that two people have the same birthday is more than 99.9%. The birthday problem is just one example where math can show that things that seem impossible, like the same person winning the lottery twice, actually aren’t unlikely at all. Sometimes coincidences aren’t as coincidental as they seem.

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