In class today I invoked one of my favourite examples of a counting problem: why are the poker hands ranked the way they are? I started explaining that the important thing is structure: more valuable poker hands are rarer, because they satisfy a more rigid set of criteria.
But wait, objected one of the back-benchers. (This particular classroom extends back a fair ways compared to its width, and so there’s a stronger-than-usual stratification with the troublemakers drifting to the back.) Isn’t every poker hand equally likely? After all, my chances of getting a royal flush in spades are just the same as my chances of getting any other set of five named cards, right?
And that is right, as far as it goes.
The thing about the poker hands specifically isn’t that any particular hand is vanishingly rare (there’s not quite 2.6 million such hands, all of which are equally likely). It’s that, a priori, we agree on interesting constraints/features to look for in a hand: cards of the same rank. Cards of the same suit. Cards in sequence. By defining these things as “interesting”, we are creating structure in the world, and imputing an order to the different hands of five-card poker.
Is there anything particularly special about the constraints of poker? No, not really, except that they’re easy to check for. One could, if one wanted, invent a game just like poker but with an entirely different set of criteria for what makes a hand interesting, but it would probably be harder to play.
This same principle applies nearly anywhere that we talk about order and chaos. There’s nothing intrinsically disorderly about having a pile of papers on a desk, in that one could imagine a set of protocols where an orderly workspace required such a thing. Any sense of order follows from the observance of preconceived constraints; the simpler the constraints are, the more “natural” the order seems. A state of disorder is only distinguished in that small changes are less noticeable: it’s easy to break from order by accident, but hard to find order (i./e. break from disorder) by accident.