Improbable Things, A Big World and the Birthday Problem

It’s a big planet and so a lot of weird things that shouldn’t happen, happen all the time. As I like to remind people, on a planet with 7.5 billion people, a billion-to-one longshot potentially happens seven-ish times a day. That can take a lot of the fun out of calling things “billion-to-one longshots”, admittedly, but that’s the way it goes on a big planet.

But this insight gets ignored all the time. To put it somewhat technically, if an event has a non-zero probability of happening, and you give it enough opportunities to happen, it will, as “enough” →  ∞,  happen. The second point is crucial, of course. You have to give the non-zero probability event enough opportunities to happen for it to happen. It isn’t enough to say, “Yo, there are seven-point-five billion of you people out there—mad stuff: happen”. The probability has to be non-zero, and the opportunities legion. 

Let’s make this concrete. Say there is a billion-to-one chance that any given human breath contains a spangly, small unicorn that can recite the complete works of Jorge Luis Borges in the original Spanish.  Given that the average person breathes around 20,000 times a day, and there are 7.5-billion people on earth, our criteria are met: we have a non-zero probability event, and  150 trillion chances a day for Borges-reciting-spangly-small-unicorns to issue forth, almost certainly surprisingly, from someone. Doing the math, we might reasonably expect 150,000 Borgesicorns a day to appear on earth. That we do not suggests we may have the probabilities wrong, but that is a separate issue. 

Sometimes we get the probabilities wrong, like with the breath-born Borgesicorns. Sometimes, however, we get the opportunities for improbable events to happen wrong. Usually, narcissists that we are, we err on the side of not realizing how many chances there are every day for nutty things to happen, what with other humans being myriad & real, and not just flitting into and out of existence when we notice them. 

The birthday problem is a good example of how humans mess this up. It is usually posed this way: How many people do you need in a room before there is a 50% chance that two of them share the same birthday? Most people screw this up by, at least implicitly, only considering their own birthday, and thinking you’d need a decent number of people to have a  50% chance of matching that single birthday. 

But that’s not what’s happening here. It isn’t how many people it takes to match a single birthday; it’s how many people it takes to match any birthday.  I’ll leave the math as an exercise, and it’s not hard, but say there are 23 people in a room, then there are 253 possible pairwise birthday comparisons. Having one of those match is unlikely— 1 in 364—but having no matches out of 253 comparisons is much less likely, slightly less than 50%. Turning that around, there is a slightly better than 50% chance in a group of 23 people that two will share the same birthday.

This result usually surprises people, and that’s great. We all like to be surprised by counterintuitive mathy things, or at least I do. But the reason why I like the birthday problem is because it’s emblematic of how imagination fails us when we are forced to think about how many opportunities there are on a crowded, connected world for strange things to happen. The next time something strange happens, whether it’s interpersonal, physical, medical, or anything else, ask yourself: Is this really that unlikely, or is it just my inability to imagine, given a non-zero probability and  x →  ∞, how wildly probable improbable things are.