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.
READER COMMENTS
Henry
Jul 18 2025 at 11:03am
My probability professor illustrated this by finding a student in the class who played the lottery and asking them if they would ever choose the numbers “111 111 111 1”.
Prof: Why not?
Student: Because it’s never going to happen
Prof: Exactly.
Robert EV
Jul 18 2025 at 5:10pm
I wouldn’t choose those numbers either, because the odds of hitting based on personal choice of numbers or machine choice is equal. So personal choice increases the transaction cost.
A real benefit of playing the lottery is that it provides far most fantasizing entertainment value than going to a movie, at a cheaper price.
john hare
Jul 19 2025 at 4:36am
That’s my thought. For the $2.00 I spend on a powerball ticket, I get to think about what I would do with $100M net. I only spend that $2.00 per draw when it is over $200M. Some have criticized this as even the single digit millions would change my life. To me, the odds are so bad that only the big dreams are worth the $2.00. I probably spend about $100.00 a year on this particular fantasy. It’s on par with the money I will spend on a steak dinner when similar nutritional value is in the hamburger steak at half the price.
Knut P. Heen
Jul 21 2025 at 9:41am
I think it was in Bulgaria they drew the same numbers two weeks in a row. This has to happen sometime, somewhere in the world.
Robert EV
Jul 18 2025 at 3:29pm
The odds of someone being gobsmackingly surprised by a straight set of spades in a game of spades is 1 in 635 billion divided by 4.
I was once playing poker with my half cousins and we were all dealt royal flushes. My half cousin realized it was because he had dealt out of a euchre deck (that presumably hadn’t been shuffled).
Our minds are trained on a basis of salience.
David Seltzer
Jul 18 2025 at 5:43pm
Kevin: Really interesting. I remember another example, flipping a fair coin. The experiment:
Flip a fair coin 1000 times in a row. If ten heads in a row come up, one might conclude the coin is not fair even though each trial is independent of the previous trial. If the coin is fair, we’ll know by the 1000th trial wherein we observe 501 heads and 499 tails,. The P( 10H in a row) =(1/2)^10. About .000976563. In the limit, as n approaches, the number of heads will match the number of tails. If one is betting a dollar on each trial, the expected payoff is zero.
Jon Leonard
Jul 18 2025 at 6:08pm
A number of these are easier to process if you realize that we’re interested in classes of hands (or events, etc.) So “A hand that is this good” is more useful than “this specific hand”, and classes in that sense are not equally likely; often not even close. But yeah, people are still bad at probability.
Mactoul
Jul 20 2025 at 12:30am
William M Briggs, the author of Uncertainty:the soul of modeling, probability and statistics is fond of saying that events do not have probabilities. Probability is epistemic in the sense of the odds of an event in the light of knowledge one already has.
Thus, the contradiction between your intuition that many 13 card sets are should not be improbable and the mathematical idea that all sets are equally improbable in themselves, this apparent contradiction dissolves.
Knut P. Heen
Jul 21 2025 at 9:35am
In one sense you are right, but not in the gamer sense. The point is that there are many average hands, but just one best hand. The probability of a particular average hand is the same as the probability of the best hand, but gamers don’t care about what particular average hand they get. They care about average vs. great hand. The probability of getting an average hand is much greater than the probability of getting the best hand possible.