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Numberplay: Rare Coincidences Are Very Common!

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“Wow!” “Amazing!” “Unbelievable!” “What are the chances of that?” Most, if not all of you, have uttered words like this at some time in your life. The paradoxical title of today’s Numberplay, then, is true: rare coincidences are really common.

Why should this be? After all, rare should be rare, shouldn’t it? People who are prone to magical thinking seize on such commonly experienced rare coincidences and ascribe cosmic significance to them, invoking Divine Providence or Pre-arranged Destiny or Synchronicity or some other favored pseudoscientific explanation. But if these coincidences are so common as to happen to everyone, then how significant can they be? It’s like that pearl of wisdom that I first heard from a treasured friend, The Talking Moose, on an old Mac computer over 20 years ago: “Remember that you are a unique individual — just like everyone else.”

Today, we’ll see how the commonness of rare coincidences can be fully explained by nothing more than an interaction of mathematics and human psychology, creating a few distinct patterns of fallacious thinking, which I’ll give as label and problem.1 The first one is Too Many Targets.

1. Suppose there are a thousand possible rare events, each of which can happen with the “unbelievable” probability of one in a million on a particular day. How likely is it that you will encounter one such unbelievable event in a year? In a lifetime of 80 years? How many such events will an average person see in his or her 80-year lifespan?

In fact, the number of possible events that we would consider “rare” are much much more than a thousand. There are so many of them that it would be fruitless to try and list all of them before they happen, and so we don’t think about them. We just reserve our oohs and aahs for the ones that do happen. Gary’s graphic says it all. You throw a dart at random on to a dark wall, and shine a flashlight to see where it went. Wow, it hit a bull’s eye! Amazing! Very few of us bother to turn on the light switch. The flashlight of our selective attention blinds us to the underlying reality. It turns out that the whole wall is tiled with bull’s eyes! Moreover, throughout your life you take thousands of potshots at them.
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