Trivia 080: A collectivized birthday

The other day I asked one of my new roommates what day their birthday was, so that I could mark it on our calendar. To my great surprise, his birthday is exactly the same as mine! What a coincidence! We just need to determine what time we were each born so that we know who is senior over the other.

Although having the same birthday as someone else was surprising (it has never happened to me before that I can recall), of course it’s bound to happen sooner or later, especially in a large enough group. But how large is large enough? The answer often surprises people.

As it turns out, you only need 23 people in a group before the odds that two of them have the same birthday is greater than 50%. Although this seems like a shockingly low number, most of the people I have talked to about this have realized that they had a class in school where two classmates had the same birthday. So it’s not as unusual as it first seems. But still, it seems like that number is way too low. After all, there are 365 days in a year (ignoring leap years – that’s next week’s topic). So how is it possible that you only need 23 people to make it a coin flip whether or not two people have the same birthday?

The answer lies in the fact that we aren’t actually interested in the number of people, but the number of pairs of people you can have. Because each unique pair of people is an opportunity for there to be a shared birthday. And you can pair 23 people up in 253 different ways. This pushes the probability of two people sharing a birthday up to ~50.7%. If you have a group of 70, the number of pairs you can create rises to a whopping 2415. In that case, the probability of a shared birthday rises all the way to 99.9%. Now, of course, keep in mind that these are not the odds that any two given people share a birthday – those are still very low, 1/365 or 0.0027% – but the odds that within the group at least two people have the same birthday. TED-Ed did a very good video explaining the mathematics behind the calculations.

This question is known as the “birthday problem“, and it’s a good example of how human intuition works well with linear extrapolations, but fails us when it comes to higher functions, like quadratics and cubics. But such relationships are all around us. For example, which is more dangerous – a collision at 25mph or at 50mph? Most people would correctly point out that a faster collision is obviously more dangerous. But how much more dangerous is it? Well, you’ve doubled your speed, so it’s doubly dangerous, right? In this case, it would actually be four times more dangerous. That’s because the amount of kinetic energy an object has is proportional to the square of its speed. (Of course, this is a very simplistic, physics textbook-esque view of things). It’s important to keep in mind that sometimes our intuition needs a bit of training.

So remember, it’s not your birthday or my birthday. It’s our birthday.

If you want to check out which famous personalities share your birthday, you can use this website. And if you’re curious to see how far around the sun you have traveled since you were born, you can check that here.

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