The High Probability of Coincidence

Coincidences are quite likely.

The odds of any particular coincidence will be quite small. But there are so many possible coincidences that the odds of any coincidence are very high.

Take any deck of cards. Shuffle it, ideally 7 times*. What are the odds that you find a pair of queens on the top of the deck? About half a percent, it turns out. Very small.

CardShuffleBut what are the odds of finding a pair of any kind, anywhere in the deck? I don’t know actually. That’s hard to calculate. But it’s much, much larger.  Go ahead, check for yourself. I just tried it five different times myself, and never failed to find at least one pair. Often more than one.

This is probably fairly obvious. But you’d be surprised how easy it is to lose sight of it.

Before a presidential election, for example, news magazines tend to fill up with stories about weird things that can predict the results. They’re played for laughs, of course. But when you see these articles discussed in comments, you’ll find lots of readers determined to explain “why” these coincidences work.

Some of them may have explanations, it’s true.** But the point is, these coincidences don’t need explaining because their existence is not unlikely.

Consider, for example, this list of “bellwether” counties which have voted with the nation every year since 1956. A lot of effort can, and has, been put into determining what makes them special. It seems to require an explanation because the odds of any particular county getting all those elections right are roughly one in a thousand. But there are 3,140 counties in the US! It’s bound to happen in at least a few places.***

The same goes for the infamous “Redskin Rule.” Surely some sports team will have won games in a manner that corresponds to election outcomes; there are so many of them out there playing so many games! To illustrate the point, Chris Wilson at Yahoo has come up with a foolproof rule for every NFL team. There are plenty of coincidences out there. We notice the ones which seem to make the most “sense” and then marvel at their seeming improbability.

It’s usually harmless fun. But this is the same kind of thinking that leads to superstition and paranoia. People pore over over lists of numerological coincidences from famous tragedies and see either cosmic plans or government conspiracies. They find patterns around the negative events in their lives and begin to believe in “bad omens” (world leaders do it too). They hear about men (or animals!) with weird systems for picking sports results, and they bet very real money thinking they’ve gotten a hot tip.****

It’s not an act of ignorance– quite the opposite. It’s a deep-rooted human impulse to try to apply our amazing mental faculties to every situation, and discern every possible pattern.

But we can’t be shocked when we encounter these oddities. There are just too many possible coincidences for us to avoid them all.




*That’s the minimum number mathematically required for a maximally random deck

**Famously, for example, Scholastic News has polled schoolchildren and gotten things right 15 out of 17 times, and that could well be a result of the fact that most young children probably “vote” just the way their parents will.

*** In describing the odds, of course, I’m assuming that all election results are independent, and that each one comes down to a coin flip. I’m also assuming that counties all vote randomly and independently. Manifestly, none of these things are true. But they’re actually not bad approximations–voting patterns change so dramatically across the years that the results Queens County in 1996 and Queens County in 1964 really might as well be independent.

Now, there certainly are some subtle correlations between counties and throughout the years which break this randomness, and I don’t mean to suggest that the careful political analysis which goes into trying to detect these is not worthwhile. But it’s very easy to overestimate the importance of any particular trend.

**** This, incidentally, is a rather famous con game. A scam artist sends a “prediction” to everyone in Denver, telling half of them that they predict the Broncos will lose, and the other half that they will win. After that week, half the town will think they called the game correctly. The next week, they further subdivide this half, sending out both predictions. Every week, some portion of the population believes that the scammer has been getting games right all season long. After 10 weeks this will only be about 1/1000th of the population. But in Denver, that still amounts to 260 households. A few of them will be gamblers convinced the man is a genius, and will gladly pay handsomely for his week 11 pick .


About Colin West
Colin West is a graduate student in quantum information theory, working at the Yang Institute for Theoretical Physics at Stony Brook University. Originally from Colorado (where he attended college), his interests outside of physics include politics, paper-folding, puzzles, playing-cards, and apparently, plosives.

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