Gary Antonick

“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.

In his book “Unweaving the Rainbow” Richard Dawkins coined a technical term for the target-studded wall of potential coincidences. He calls it a PETWHAC — “Population of Events That Would Have Appeared Coincidental.” In most circumstances, the PETWHAC is huge. After all, in today’s Age of Information age we are bombarded every day with thousands of facts, names and numbers that can become fodder for coincidence.

Sometimes you don’t find coincidence, coincidence finds you. This brings us to our second label. It’s called You Are The Chosen.

2. Imagine that you are following a series of eight important ballgames. The day before each ballgame, you receive an e-mail accurately naming the team that is going to win. It’s uncanny! The e-mail admits that similar predictions have been sent out to other people, but that each prediction is “personalized” and is solely dependent on the receiver’s psychic persona. Believing that you are attracting good luck, you willingly part with a small amount of money, say $10, to be tipped off about the next game. After all, you stand to win a lot more with the bookmakers. Assuming the games are fair with each team equally likely to win, and the scammers are guessing too, how could such a scam be carried out? How many people will they have to e-mail to earn $500 on the ninth game if half of the eligible people fall for the bait? How many more games on average after the eighth before the e-mail oracle fails? 3. The above problem had a real world counterpart in the recent World Cup. An octopus named Paul correctly predicted the results of each of the eight games that Germany played. What do you think is the most likely explanation for Paul’s success?

There are many other fallacious reasons why coincidences seem so common. Among them are Overestimating the Odds, Making A Good Story, Parallel Thinking, In-Step Emotional Time Constants, Subconscious Cues, Publicity Seeking and of course, the age-old favorite – There’s A Sucker Born Every Minute. I won’t dilate on them here, but you can try and guess what they might be. If this topic is a hit, I may give you more puzzles based on it next time.

I’d love to hear about examples of coincidences in your own experience and the possible or real reasons for them. You can also try to come up with oxymorons such as the title of today’s Numberplay and the sage utterance of the Talking Moose.

In parting, I’d like to put forth what I think the proper emotional attitude to such coincidences should be. Do rational explanations of coincidences take the magic out of life? Do they, in fact, “unweave the rainbow,” as the poet Keats accused the scientist Newton of doing, in a diatribe that inspired the title of Dawkins’s book? Not at all! If you are anything like me, you have been as moved and have shed as many tears watching and reading fictitious stories as you have in real life, even though you knew that the stories weren’t true. Enjoying the wonder of something has nothing to do with intellectually believing that it is true or cosmically significant.

So I say, let’s enjoy the coincidences that come our way. It’s an emotional response that’s wired into our brains, a throwback to the times when we lived in small groups and encountered only a small fraction of the information we do today. Be liberal with “Wow!”and “Amazing!” and “Unbelievable!” Enjoy them to the fullest. Just don’t take the coincidences seriously. They usually mean absolutely nothing.

May your coming week be blessed with many interesting coincidences. And may you not assign cosmic significance to them.

1. That’s my version of “chapter and verse.”

Well-explained and correct solutions to the questions posed in this post will be highlighted in a couple of weeks and referenced and discussed in the subsequent Numberplay column.