The Birthday Problem

Look around. Do you see any people? If you answered yes, then please proceed. If not, then go find some friends. Found some? Good. Do they all have birthdays? Great, we’ll need those. There’s an interesting “paradox” in probability theory called the “Birthday Problem”, and I’m going to tell you about it. If you go ask your friends what their birthdays are, intuition usually says that the chance for any two of them to have the same birthday is small. After all, there are roughly 365 days in a year, and assuming we’re equally likely to be born on any given day, that means we have quite a lot of “choice”. However, if you now record all your friends’ birthdays, you’ll be surprised to see that two of them will probably share birthdays (depending on the number of friends, of course). In fact, if you have about 23 friends, there’s roughly a fifty-fifty chance that at least two of them will share birthdays, which is totally counterintuitive. What’s wrong then? Could it be that we don’t have that much “choice” after all? Well, not quite. The problem here lies in our intuition, which turned out to be wrong. If we think of how likely it is for two people (at least) in a group to share a birthday, we’re very likely to do something wrong, and then get the wrong answer. However, luckily for us, there’s an easier way to think about it: instead of how likely it is to happen, let’s think of how unlikely it is to happen. Imagine the following situation: a person enters a room, and they have a birthday (let’s not include leap years for now). What’s the chance they won’t share a birthday with anyone in the room? Well, it’s impossible, since they share a birthday with themselves. A second person enters the room. In order for them to not share a birthday with anyone else in the room (i.e. the one other person), they need to “choose” from 364 days. So the chance for two people to not share a birthday is 364/365. Now a third person enters the room, and they now need to choose from 363 days. You can probably guess where this is going, so I’ll leave the rest as an exercise. The formula you end up getting is $\frac{^{365}P_N}{365^N}$ , where N is the number of people in your group. Now, all you have left to do is convert this result into the thing you want: the probability of people sharing birthdays. That’s easy though, since you just found the formula of its complementary probability, so you just subtract the formula from 1. And voilà, you get what you wanted, neatly expressed as $1-\frac{^{365}P_N}{365^N}$ . Graphing this yields the following:

I’m only showing you a small chunk of the function here, since it gets quite boring after 80 people; at about 70 people, the probability becomes 99.9%, and it doesn’t change much from that point onwards.

Hopefully you learned something new from this little thought exercise: either that our intuition is not always right, or that you need to go find some new friends.