Coincidence!

I've only recently started this intermittent blog, and a few days ago I sent off an email to a friend, notifying him of its existence. We haven't communicated for a few months, so it was amusing to get back the response:

(interesting stuff about stuff ....)
Hey, I was in the middle of this when YOUR e-mail came in! Telepathy?

Now, I know he doesn't really think it's telepathy, but some people do get exercised about this sort of coincidence. But things like this happen all the time. And it's easy to see why.

It might seem like the question is: "What's the chance of me receiving an email from you just as I'm writing one to you, given we haven't emailed each other for ages?" Pretty small, I suspect. But actually, the question is really: "What's the chance of me receiving an email from you just as I'm writing one to you, given we haven't emailed each other for ages, and given that I've just received an email from you just as I'm writing you one?". That is, what is p(x|x) (the probability of x having happened, given that x has happened)? Well, it's one. You can't get less unlikely than that!

Okay, that seems a little unsatisfactory. It still seems somehow to be remarkably unlikely. What's the probability it will happen again? Very small. But what's the probability that some weird coincidence will happen again? Now that is rather high.

Let's assume that we think some event has a probability of one in a gazillion of happening (the probability prior to it actually having happened, that is). But there are equally gazillions of unlikely things that could happen. Say you get an email from a friend just as you were thinking of them. But they might have phoned you, or texted you, or visited you, or written to you. Or you might have seen them on TV, or read about them, or about someone with the same name. And you could have been thinking of any of your friends, or of anyone else, or of anything else.
There are oodles of possible unlikely coincidences. What are the odds that one of them happens?

The way probability works, it's easier to calculate the chance of none of them happening. Let's say the odds of each one of these things happening is one in N, where N is very large (one in a billion, one in a trillion, or more). So the probability is 1/N, and the probability of it not happening is 1-1/N (very nearly, but not quite, certain that it won't happen).

Now let's say the number of unlikely things that might happen is also this huge number N (it could be 10N, or N/10; the calculation is cleaner using N, but the overall flavour of the result still holds for other values).What is the probability that none of the N unlikely things happens? That is, what is the probability that the first thing doesn't happen, and the second thing doesn't happen, and ... all the way up to and the Nth thing doesn't happen?

We just multiply the individual probabilities together, so we get (1-1/N)^N. That's the probability of no coincidences, so the probability of at least one coincidence is p = 1-(1-1/N)^N. For N larger that about 100, p is about 63% (for 10N it's 99.99%, for N/10 it's 10%). That's a pretty good chance of a weird coincidence. And that's just today!

The moral is: when individual events are unlikely, but there are also a lot of events that could happen, something will almost certainly occur.

So, that email: unlikely coincidence? That coincidence, yes; some sort of coincidence happening, not really.