JREF Swift Blog

Understanding Coincidence

Many people have a legitimate fear of numbers, equations, and probability. This “math anxiety” keeps much of the lay public from ever willfully learning about mathematics; indeed, ignorance in this regard is often touted. Commonly used phrases like “I’m not a numbers person” and “I hate math” betray that fact that a good portion of society does not understand math and consciously avoids it.

Comprehending this deficit and doing something about it should be taken up within our school system; we should engage students with math early, often, and more rigorously.

But mathematical illiteracy plays a role in perpetuating not just equation ignorance, but pseudoscience. Not understanding just how much of your life is governed by randomness generates many a fallacious belief about the way that the world works. It should be clearly understood that randomness creates coincidence. That is to say, if there were no coincidences in life, we could speculate that some outside force is controlling the events in our lives. However, with true randomness comes the expectation that coincidences will happen: there will be cancer clusters, your friend will call you just when you were thinking about them, and last night’s dream will have somehow “predicted” the events of the following day. It is with the last example, predictive dreams, which I would like to press on with. With a short lesson in randomness and probability, we can see that so-called predictive dreams (and any other event “too amazing to be a coincidence”) are nothing more than random happenings. You don’t have ESP, it’s not fate, and it’s not magic.

“I Dreamt This Would Happen!”

The purpose of this example is to show that many pseudoscientific ideas about the way the universe works are driven by a misunderstanding of randomness and probability. While predictive dreams are harmless, I would suspect that this belief characterizes the kind of thinking that underlies pseudosciences like astrology, ESP, and parapsychology.

Let’s overcome our math anxiety with a dreaded word problem. Let’s stipulate that the chance of a dream to some extent matching the events of the following day is 1 in 10,000. This means that out of 10,000 dreams, the vast majority, 9,999, will not match any future events. Let’s also assume that having a non-matching dream one night will not affect the dream of the next night, so each night is independent from one another. So given these stipulations, the odds of having a dream that does not match any real life event is 9,999/10,000. When people speak about predictive dreams, it is not as though they have them every night. If this were happening, we might consider it to be more than coincidence. However, anyone who has experienced this phenomenon (myself included) will probably tell you that they do not hit a homerun every night. It is this fact, that an amazingly serendipitous event only happens once in a while, that alludes to chance as the rational explanation.

Remembering the odds above, the chance of having a dream that does not match any real life event for two nights in a row will follow the multiplication principle of probabilities, meaning that the probability is (9,999/10,000)*(9,999/10,000). Likewise, the probability that you will have a dream that does not predict anything for three nights in a row is (9,999/10,000)*(9,999/10,000)*(9,999/10,000). Following this principle, the chance that you will have successive dreams that do not match reality can be expressed as (9,999/10,000)N, where N is the number of nights. As I said above, I don’t think that anyone would say that these predictions are a common occurrence, so let’s consider a time period of one year. The probability that you will have successive dreams every night for a year that do not predict anything would be (9,999/10,0009)365, with N equal to the number of days in a year. This results in a 96.4 percent chance that people who dream every night of a year with not have any predictive dreams. This of course means that over a period of one year, 3.6% of people who dream every night will have at least one dream that matches reality in some way. Consider that for a moment. Even though coincidences like these can drive people to believe in fate, precognition, ESP, etc., using our definition here we can say that these probabilities in large population would produce literally millions of predictive dreams each year! Even if we relax our standards and make a predictive dream a one-in-a-million event, it would still produce thousands upon thousands of predictive dreams each year by chance alone.

It’s not magic, it’s not fate, it’s not a spiritual connection with someone else; if there’s a likelihood that something will happen, however small, it is explained by chance alone that it is bound to happen to some people at some time. Look at what happened with the supposedly prophetic Nostradamus. He threw out a claim that had to do with two towers coming down and hundreds of years later something similar happened. Somehow this passes for incredible predictive powers. Knowing what you now know about randomness and large numbers, what do you think the chances are that if I through out a vague claim, something similar to that claim will happen in the next 500 years? Would that be evidence of a magical precognition on my part, or is it just randomness? The same goes for the pseudoscientists who claim to predict natural disasters. Don’t make them famous for saying that they predicted a major earthquake to happen somewhere, there’s a probability that they will guess correctly by chance alone. If they throw out enough random predictions, the statistics say that it is bound to match a few times.

What are the Odds?

Of course, saying there are odds that you will “predict” some event tomorrow with your dream tonight does not deal with the fact that you seemingly foresaw something. But this too can be dealt with. Consider the fact that what you “see” in your more bizarre dreams never comes true. You never see a great white shark with braces or are suddenly chased down the street by a cheeseburger. Only the things that are already within the realm of possibility have a possibility of happening. Let’s say that you have a dream that your friend will get in a car accident. Given how many car accidents occur, this argument falls to the same reasoning that we dealt with above: in a large population, statistically unlikely events will happen all the time (and car accidents are much more likely than you may think). Fine then, what if you dream that you will finally get that promotion tomorrow specifically at 4:30 PM and it happens? It may seem amazing on its face, but considering that you were most likely already in line for that promotion, have been thinking about it a lot, and that many employers make calls like this near the end of the work day, it is not amazing at all. If you also acknowledge that those who believe in the “power” of predictive dreams are actively seeking events in their day to match their dreams and reinforce their belief, we realize that we have purely random coincidences with a pinch of confirmation bias, nothing more. This active seeking combined with randomness is a death knell for many pseudosciences that rely on math illiteracy (i.e. astrology can be rationally explained away in this view).

It boils down to this: unlikely events happen all the time because we live in a random world filled with billions of people. Even when you seemingly predict an event with a dream, we would expect some matches from chance alone. Also, you only predict events that have some prior probability of happening anyway. When you line all of the various probabilities up, a "miraculous" event is bound to happen to someone at sometime. Even one-in-a-trillion events happen to people, it’s just a matter of probability.

If “amazing” coincidences like predictive dreams did not ever happen, we could be suspicious about the workings of our world. However, this is not what we see. We see and expect supremely unlikely coincidences to happen through chance alone. Understanding these facts is the first step to thinking probabilistically and therefore more accurately, inoculating you from some of the pseudosciences that prey on these deficiencies. Overcoming your math anxiety will make you a better critical thinker, and trust me, there’s a good chance of that.

Examples from this post were adapted from the book “Innumeracy” by John Allen Paulos.

Kyle Hill is the newly appointed JREF research fellow specializing in communication research and human information processing. He writes daily at the Science-Based Life blog and you can follow him on Twitter here.