Is the Birthday Paradox Really a Paradox?
How many people would you have to gather together in a room to ensure at least a 50% chance that two people in the room share a birthday? Some people might think you’d need 183 people, since that’s half of 366. But they would be wrong! Would you believe you only need 23 people? It seems impossible, but it’s true! This interesting mathematical oddity is known as the birthday paradox.
Of course, it’s not a true logical paradox, because it’s not self-contradictory. It’s just much unexpected and surprises most people, so it seems like a paradox.
How does the math work? Before we get started on that, let’s assume from here on out that there are only 365 possible birthdays and that every birthday is equally likely. While those assumptions aren’t completely accurate, they make the math easier and don’t affect the results in any meaningful way.
The birthday paradox is so surprising because we usually tend to view such problems from our own perspective. For example, if you walk into a room with 22 other people, the chances are pretty good that no one else will have the same birthday as you. With only 22 of the possible 365 days taken up, that leaves 343 out of 365 chances that your birthday will be unique.
Only considering things from our own perspective, however, limits our expectations. Instead of making 22 comparisons (our own birthday versus the other 22 people in the room), we have to compare each person’s birthday to every other person’s birthday in the room.
The first person compares with 22 other people. The second person compares with 21 other people (subtracting one since the first person already compared with the second).
The third person makes 20 comparisons and so on, down to the second-to-last person only comparing with one other person, the last person. Adding up all these comparisons among 23 people (22 + 21 + 20…+ 1) gives us a total of 253 possible pairs, which makes it much more likely that we’ll find a pair with matching birthdays.
Without diving too deeply into complex probability calculations, let’s take a look at the probability that, in a room of 23 people, no one has the same birthday as another person. Experts say that’s the easier calculation to make.
The probability that person 1 has a unique birthday is 365/365 since every date is available. For person 2, the probability drops to 364/365, since one date is taken by person 1.
That trend continues until we get to person 23, whose probability of having a unique birthday is 343/365. We must multiply all 23 separate probabilities to find out the probability of everyone having unique birthdays. Doing the math, we would end up with a probability of 0.491.
Logic tells us that subtracting our result from 1 will give us the probability that at least two people out of the 23 share a birthday. That means that 1 – 0.491, or 0.509 or 50.9%, is the probability that at least two people in the group of 23 share a birthday.
Adding people to the room will increase the probability that at least one pair of people share a birthday. For example, in a classroom of 30 students, you’d have a 70% chance of two classmates sharing a birthday. If you increase the number of people in the room to 70, there’s a 99.9% chance that a pair of people will have the same birthday!