The probability of coincidence


I have a series of posts coming up about the probabilities attached to the “out of left field” events that hit your life like cancer, how to understand the statistics, and how we have attached various personal “theodices” (the “justice of God”) to these events. Some of that is a bit of downer, so I thought I would post a brief look at the fun side of this type of occurrence, which often hits us in the form of unexpected coincidences.

So, you are driving down the road and you fix your eyes on an interesting vanity license plate on the car in front of you. Let’s say it reads, “SUZIE-Q”. And not long after seeing that license plate, you hear the old Creedence Clearwater Revival hit song “Suzie Q” playing on the radio. And you say to yourself, “Now that’s a coincidence!”

The late physicist Richard Feynman was known as a bit of a joker, as much as physicists can be jokers, and this is a joke that only mathematicians and physicists would probably find amusing. He started out a public lecture telling this story:

“You know, the most amazing thing happened to me tonight. I was coming here, on the way to the lecture, and I came in through the parking lot. And you won’t believe what happened. I saw a car with the license plate ARW 357. Can you imagine? Of all the millions of license plates in the state, what was the chance that I would see that particular one tonight? Amazing!” [1]

The joke here is that you seeing a license plate with the unremarkable tag reading “ARW 357”, or any other tag, has exactly the same probability as you seeing the tag reading “SUZIE-Q”. This is another application of Poisson’s Law of Large Numbers, with the addition of a fascinating function of our brains.

On any given day driving your car down a busy road, your eyes are capturing, and your brain’s visual cortex is processing, hundreds of license plates, but usually subconsciously, as well as thousands of other observations about your changing surroundings. Behind the scenes your brain neurons are doing pattern matching, tossing out most of the thousands of pieces of visual input coming in until it scores a “hit.” As neuroscientist Daniel Dennett describes it:

“The task of the mind is to produce future, as the poet Paul Valery once put it. A mind is fundamentally an anticipator, an expectation-generator. It mines the present for clues, which it refines with the help of the materials it has saved from the past, turning them into anticipations of the future. And then it acts, rationally, on the basis of those hard-won anticipations.” [2]

So, our brains are always doing this “data mining,” looking for potential connections, but we rarely consciously perceive that process of “expectation generating.” My preferred way to look at our nightly dreams, which is also one of the classic theories, is to see them as “garbage processing,” where our brains are trying out a lot of random associations. My brain is comparing the day’s many thousands of perceived events, stored in short-term memory, to memories stored long-term, in order to figure out what is worth saving and what can be “trashed.”

Likewise, the sounds we are always listening to. If you tend to listen to an “Oldies” radio station in your car, your brain is also processing every song subconsciously your stored memories. So even with a “lottery-level” probability that you might hear “Suzie Q” on the radio the same day you see the license plate, your mind has processed millions of “lottery ticket” associations where “no important associations” clicked, and the input was discarded.

An evolutionary psychologist would see this as a basic mammalian survival mechanism. Our brains are good at processing random associations, looking for important patterns indicating food, physical threat or reproduction potential. In the process it makes a lot of “false positive” hits, but better this than missing the pattern of a lion lying in wait in the underbrush. Humans who process these patterns more effectively than others are more likely to be around when it comes time to reproduce, which was the point of a more recent post.

We also tend to greatly underestimate the number of “lottery ticket” associations our brains are making every day. A great example of this is the “Six Degrees of Separation” game, where you are able to “connect” one person to another through a chain of six or fewer other people or events that link one to the other.

The numbers of connections that our brains can potentially process in this game is staggering. Even if we know just 200 people, for instance, and we assume the same for everybody else we meet, just one degree of separation might include a pool of 40,000 people (200 of my acquaintances times 200 for each of those 200 acquaintances), and then two degrees of separation can encompass 8 million people (200 times 200 times 200)! There are certain to be a lot of overlaps in that set of people, but then you are also likely to be acquainted with more than 200 people if you try to count them.

And thus, you very likely “know somebody who knows somebody who knows somebody.” It’s not a coincidence. It’s math. And if it were not “Suzie Q” that your brain flagged today, it would most likely be something else that triggers your “Wow, what a coincidence!” response.


  1. David L. Goodstein, “Richard P. Feynman, Teacher,” Physics Today, volume 42, number 2 (February 1989), p. 73.
  2. Dennett, Daniel C. Kinds of Minds: toward an Understanding of Consciousness. Basic Books, 1998, p. 57.

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2 thoughts on “The probability of coincidence

  1. Pingback: You are a lottery winner! – When God Plays Dice

  2. Pingback: Cancer, probability, normality and theodicy – part 1 – When God Plays Dice

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