#1242 What is the birthday paradox?

What is the birthday paradox? It is the fact that you only need 23 people in a room for the chance of two of them having the same birthday being over 50%. And it’s not a paradox.

This probability problem is interesting and it is very hard for us to understand for several reasons. It is also not actually a paradox. A paradox is, according to the Cambridge dictionary “a situation or statement that seems impossible or is difficult to understand because it contains two opposite facts or characteristics.” The Fermi Paradox is a good example of a paradox. The Fermi paradox was first voiced by Enrico Fermi. He said that there is a paradox because there is no evidence of intelligent extraterrestrial life and yet the universe is so massive that the chance of intelligent life existing is almost 100%. There are several reasons for this, which you can see here, if you are interested. Anyway, that is a paradox because there are two contradictory facts: intelligent life doesn’t exist vs intelligent life must exist. The birthday paradox, on the other hand, is not a paradox, it is just difficult to understand. Let’s have a look at it and why it is hard to understand.

The question is, “how many people do you need to get together before the chance of two of them having the same birthday is more than 50%” The answer is 23 people. If you have 70 people or more, you will have a 99.9% chance that two of them will share a birthday. Why do we find this hard to understand? Intuitively, we might think that there are 365 days in the year, so we would need half of that, which is 183ish to have a 50% chance that two of them will have a birthday. Some people might think the number is even higher because they might think that the birthday has to be the same as theirs. Or that the birthday must be the same as person A’s. These are all incorrect because we don’t think about all the possible pairings that don’t contain us, or person A. Then, when we are told the answer is 23, we find it very hard to understand. We hear 23 and know that there are 365 days in a year, so the number seems too small. This is because we think 23 only gives 23 possible pairings, but it doesn’t.

The trick to understanding this is to think about how many possible pairings you can get and how many pairings you need to have a good chance that two people will share a birthday. If we have two people, A and B, we have one possible pairing: AB. If we introduce a third person, C, we now have three possible pairings: AB, AB, BC. Here comes person D. Now we have six pairings: AB, AC, AD, BC, BC, CD. Enter person E and now we have five people. We now have ten possible pairings: AB, AC, AD, AE, BC, BD, BE, CD, CE, DE. And the number of pairings keeps increasing. With six people there are 15 possible pairings. Seven people = 21 pairings. It keeps going up until with 23 people you have 253 possible pairings. 253 is more than half of the days in a year and if we were asked “what is the chance that there will be a shared birthday out of 253 possible pairings?” we wouldn’t be so surprised or have trouble believing it.

To work this out mathematically, it is simpler to look at the probability that the group of people won’t have the same birthday. If you have people A and B, you only have one pair so there is only one chance they will share a birthday and 364 chances they won’t. That can be written as X, which is 0.997260, or 99%. This is the chance that they won’t share a birthday. If you have three people, you get XX, which is 0.99179. Still very high but getting lower. When we have ten people, we get XXXXXXX, which is 0.883. 88% chance that ten people won’t share a birthday, but a 12% chance that they will. This keeps going until you have 23 people. XXXXXXXXXXXXXXXXXX, which is 0.4927 or 49%. Remember, this is the chance that two people in the group won’t share a birthday, which means we now have a 51% (slightly over half) chance that two of the people will share a birthday. And the odds rapidly increase as you add more people. With 30 people, you have a 70.6% chance. With 40 people it’s 89.1%. With 50 people it’s 97%. Of course, this is just probability. Even with a thousand people in a room there is still a chance that none of them will share a birthday, but that chance is so small that you would have to repeat the experiment billions of times for it to happen once. And this is what I learned today.