Are coincidences a blip in the matrix? Are they evidence that our lives and the universe have purpose and meaning? Unfortunately, no. If you like to think of coincidences as some sort of supernatural event, don’t read any further. In reality, coincidences only seem amazing due to improper framing.

Holy Cow! What Are You Doing Here?

Let’s say that you are in London on vacation and go to dinner at a little restaurant in Notting Hill. Ater being seated you look over and see that your college roommate and her husband are seated at the table next to you! Wow! What are the chances? Is there some higher power that caused this to happen? Was there a blip in the matrix? Was it fate that you’d be seated next to each other?

While it seems counter-intuitive, the highly improbable happens all the time. As summed up by the British statistician R.A. Fisher: “the ‘one chance in a million’ will undoubtedly occur, with no less and no more than its appropriate frequency, however surprised we may be that it should occur to us.” Likewise, Aristotle observed: “it is probable that improbable things will happen. Granted this, one might argue that what is improbable is probable.”

Littlewood’s Law of Miracles

Mathematician John Littlewood calculated that the average person would experience a miracle about once a month. He defined a miracle as a one-in-a-million occurrence. Physicist Freeman Dyson summarized Littlewood’s law of miracles as follows:

Littlewood’s law of miracles states that in the course of any normal person’s life, miracles happen at the rate of roughly one per month. The proof of the law is simple. During the time that we are awake and actively engaged in living our lives, roughly for eight hours each day, we see and hear things happening at a rate of one per second. So the total number of events that happen to us is about 30,000 per day, or about a million per month. With few exceptions, these events are not miracles because they are insignificant. The chance of a miracle is about one per million events. Therefore we should expect about one miracle to happen, on the average, every month.

The reason coincidences seem so rare is that we aren’t paying attention to the other million boring, unremarkable experiences that occur the rest of the month. Only the unusual stands out. This notion is captured perfectly by physicist Richard Feynman:

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!

The Highly Improbable Happens All the Time

Let’s look at this in action with an example relating from Joran Ellenberg’s book How Not To Be Wrong: The Power of Mathematical Thinking:

Most coincidences lose their snap when viewed from the appropriate distance. On July 9, 2007, the North Carolina Cash 5 lottery numbers came up 4, 21, 23, 34, 39. Two days later, the same five numbers came up again. That seems highly unlikely, and it seems that way because it is. The chance of those two lottery draws matching by pure chance was tiny, less than two in a million. But that’s not the relevant question, if you’re deciding how impressed to be. After all, the Cash 5 game had already been going on for almost a year, offering many opportunities for coincidence; it turns out the chance some three-day period would have seen two identical Cash 5 draws was a much less miraculous one in a thousand. And Cash 5 isn’t the only game in town. There are hundreds of five-number lottery games running all over the country, and have been for years; when you put them all together, it’s not at all surprising that you get a coincidence like two identical draws in three days. That doesn’t make each individual coincidence any less improbable. But here comes the chorus again: improbable things happen a lot.

ANOTHER EXAMPLE: A woman won the New Jersey lottery twice in four months. This coincidence was reported as being a 1 in 17 Trillion chance. Two Harvard professors – revising the framing as above – found that the chance that such an event could happen somewhere in the U.S. is about one in thirty!

To reiterate Dr. Ellenberg’s point: coincidences don’t seem so amazing when you apply the appropriate framing to the situation. Going back to the example of being seated next to your college roommate while on vacation in London, how should we think about this? If the question is “what are the chances that on this particular vacation to a foreign city I’ll be seated next to my roommate” then the coincidence is indeed amazing. However, if we frame the situation as “what are the chances that on one of my vacations I’ll encounter a someone I know from out of the hundreds or thousands of people I know” then the coincidence doesn’t seem so amazing. It is this latter question that is the appropriate framing.

Again from Dr. Ellenberg: “The mistake is in being surprised by this encounter with the improbable. The universe is big, and if you’re sufficiently attuned to amazingly improbable occurrences, you’ll find them. Improbable things happen a lot.”

Notwithstanding this bah humbug of an IFOD, I do love a good coincidence. One of my favorite podcast episodes is from This American Life which delves into mind-blowing coincidences. It’s a great listen. Here’s the link: No Coincidence, No Story!