Selective Coincidences

I really enjoyed Scott Sumner’s recent post about how people are bad at understanding coincidences. There are many reasons why we can be bad at this, but one I want to talk about here is that we only selectively recognize certain coincidences, making them appear far more striking than they really are.

Here’s an example of this phenomenon I frequently catch myself making. Growing up, my family developed a tradition of playing spades – usually my dad and myself on a team against my mom and younger sister. Every now and then, I would get a hand that made me think “wow – what are the odds of getting a hand like this?” And then I’d (usually) catch myself and remind myself that the odds of my getting this particular mix of 13 cards are exactly the same as any other mix of 13 cards.

Why did I have this reaction to some hands, but not others (or even most)? I’d tend to have that knee jerk reaction when I got a hand that seemed unusual in a really noticeable way and would particularly impact how many tricks I could expect to win for that hand. If I received a total of seven cards with the spades suit in my hand, that would mean my hand was unusually strong and I could pull off an above average number of tricks. Or, if every card in my hand was, say, a seven or lower, then my hand was usually weak and I might consider making a zero bid. Most hands I was dealt, though, weren’t composed in a way that made them look immediately distinctive. Most hands had an even-ish mix of black and red cards, of suits, and of card values.

To use the extreme case, consider these two possible spades hands I might be dealt:

• Hand one: Ace of Spades, Seven of Hearts, King of Clubs, Two of Diamonds, Ten of Spades, Five of Clubs, Jack of Hearts, Three of Spades, Queen of Diamonds, Nine of Spades, Six of Hearts, Eight of Clubs, Four of Spades.
• Hand two: Two of Spades, Three of Spades, Four of Spades, Five of Spades, Six of Spades, Seven of Spades, Eight of Spades, Nine of Spades, Ten of Spades, Jack of Spades, Queen of Spades, King of Spades, Ace of Spades.

If I had the first hand, I’d just glance over it and start thinking about how many tricks I should bid, but I wouldn’t give it a second thought beyond that. If I had the second hand, I’d fall out of my seat with amazement at this once-in-a-thousand-lifetimes coincidence, and nobody who played spades would ever believe it was real. Indeed, if I were playing a game with someone and they got that second hand, I’d immediately assume they had cheated (or that they were a skilled card magician, which is kind of the same thing).

And yet, each of these hands has exactly the same odds of being dealt. But the second one just feels intuitively more unlikely, because the first one basically looks like what we imagine randomness looks like, while the second does not. The strength of the first hand is within the normal range, while the second hand is invincible. That’s why I’d never take notice of the one-in-a-thousand-lifetimes coincidence of the first hand. Even though the odds of that first hand are very low (about 1 in 635 billion*), the effect of having that particular hand isn’t very noticeable. Every time you’re dealt a 13-card hand in spades, you are witnessing something that is thousands of times less likely than winning the Powerball – but that’s just the kind of coincidence we selectively overlook.

(*The total number of 13-card hands you could be dealt comes out to 52! / (13! * (52 – 13)!), leading to 635,013,559,600 possible hands.)

If I told you “The odds of X happening to you are approximately 1 in 635 billion,” you might reasonably conclude that you can be near-certain that X will never occur in your lifetime. And yet, every time you are dealt a hand in spades, something with a 1 in 635 billion chance occurs. Massively improbable occurrences happen all the time—but we mostly don’t notice them.

Even more staggeringly improbable is the arrangement of any given deck of cards you shuffle. There are 52! ways a deck of cards can be arranged—or, fully written out, there are

80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000

possible unique arrangements into which a deck of cards can be shuffled. (See this for an attempt to illustrate just how mind-meltingly huge that number is.) Every time you shuffle a deck of cards, it is all but certain you are creating an arrangement that has never existed before and will never exist again until the heat death of the universe. But even knowing that, I’ll never shuffle a deck of cards and be floored at the nearly impossible odds of the arrangement I just created—unless it looks distinctive in some way (maybe by alternating red and black cards throughout the entire arrangement).

This is all stuff I know intellectually, but still fail to grasp instinctively—hence my knee-jerk reaction to particular spades hands as if they were unusually unlikely to happen. But even though my “system one” mind has this knee jerk reaction, it’s still good train your “system two” mind to jump in and remind yourself that the mundane things around you are just as miraculously unlikely as other things that seem much more striking—and that dramatic coincidence that caught your eye maybe isn’t all it’s cracked up to be.