I found this academic paper, on a topic at the intersection of philosophy and math, a hundred times more interesting than one expects an "academic paper" to be. If a tossed coin lands heads 92 times in a row, would you be surprised? The author of the paper, Martin Smith, argues that you should not be surprised, and makes a stronger case than I for one would have thought possible.

The article, very clearly written, is only about 7 pages long, but I understand if you don't want to click and read so will try to give the main points myself in a lot less than 7 pages. We should make a distinction between what you'd predict and what would surprise you. Suppose you determined to flip a coin just five times. Would you predict that it would land heads every time? No, because most likely it wouldn't. To be precise, the chance of this is only 1/2^{5}, or 1/32, a titch above 3%. Pretty unlikely. But suppose you then flipped the coin five times and got five heads. Should you be surprised? No, because there had to be some specific result, and HHHHH is just as likely as, say, THHTH. If you're not surprised by THHTH, then you shouldn't be surprised by HHHHH, either. You wouldn't have predicted five heads in a row. Most likely, something else happens. But "something else" encompasses all the other possibilities. If you actually flip the coin five times, there has to be some specific result, and HHHHH is as likely as any other. Five heads in a row doesn't require an explanation.

Neither does 92 heads in a row, for the same reason. That is, the argument doesn't change on account of the number of trials. (Smith chose 92 because, in a Tom Stoppard play, one of the characters flips a coin in a gambling game—heads he loses—and as the curtain rises is trying to make sense out of having lost 92 games in a row.)

My reasoning faculties find the argument to be sound. All the rest of me says to keep looking for the flaw. This isn't an unusual sensation in human affairs and reminds me of a story about the paradox of Zeno, who postulated that you could never traverse the distance between two points, since you'd first have to cover half the distance, and then half the balance, and on and on, with the result that, at the end of time, you'd still be some distance away from the destination. To demonstrate, a teacher had a boy and a girl face each other from opposite ends of a classroom. The boy then advanced toward the girl, the teacher measuring off half the distance each time. After a few advances, the boy was getting pretty close, and grinned at the girl. Trusting Zeno, she said, "You'll never get here." He answered, "Right, but I'll get close enough for all *practical* purposes." Maybe he was being a jokester, maybe he sensed that there was something unfair about chopping up distance but not time.

One's strong impression of reality, or whatever you want to call it, might make you doubt Zeno, but it isn't always a good guide to the truth. Apparently it's true that all matter is made of just over a hundred different kinds of atoms—these tiny particles we can't see that jiggle about constantly, because they attract one another until, having been drawn together, they instead repel. I anyway think the atomic hypothesis is true, though it's not on account of my "strong impression of reality" that I've reached this conclusion. Crediting only my own impressions, I'd say there are many hundreds, maybe thousands, of different substances just in my house alone.

I'm inclined to give my assent to Smith, partly because it seems people's probabilistic impressions are unusually susceptible to error, so that what seems true might very well not be. There is, for example, an entire literature about "the Monty Hall problem"—so called, I think, because it's hard to make people believe that the best strategy in a certain game is indeed the best strategy. If you're of a certain age, like mine, you'll know that there used to be a game show hosted by the TV personality Monty Hall. In the big game at the end of the show, there would be three prizes concealed behind three doors. Behind two of the doors would be something like a live, impassive goat. But behind the third door there'd be something like a sports car. The contestant gets the prize behind the door she chooses. The wrinkle is that, after she's made her choice, Monty reveals the prize behind one of the unselected doors. This is always a goat, so it's evident that Monty knows where the big prize is and wants to heighten the suspense. He then offers the contestant a deal (the name of the show was "Let's Make a Deal"): if he wants, he can take the prize behind the remaining closed door instead of the one behind the door he originally chose.

Should the contestant switch? Does it make a difference? In experiments, more than 80 percent of people stick with their original choice (according to the Wikipedia article on the Monty Hall Problem). It's not that hard to show, however, that by switching a contestant doubles the chance of winning the valuable prize. After making your original choice you have a one-third chance of winning—that's not controversial. It seems people must reason that the chance they've chosen correctly rises after Monty reveals a goat. This reasoning would be sound if Monty revealed at random what was behind one of the unselected doors. Remember, though, it never happens that he reveals a sports car and the contestant immediately loses. It's always a goat, followed by the offer to switch. The likelihood that the contestant has chosen correctly therefore remains one-third. Monty is in effect divulging that the two-thirds chance the contestant was wrong is not evenly distributed between two unselected doors. Rather, all two-thirds attaches to the unselected door Monty hasn't yet opened. By switching, the chance of winning the car goes from one-third to two-thirds.

I can't think of an experimental test for Smith's argument, but, if you doubt the one I've sketched out above for the Monty Hall Problem, you might accurately simulate the game by taking a couple of deuces (goats) and an ace (sports car) from a deck of cards. After "shuffling," deal face down one card to yourself and two to someone you are co-quarantining with. Now, to make it more interesting, you each ante a dollar and the person with the ace wins the pot. This is obviously a bad game for you, right? Suppose you make a rule that, before seeing who has the ace, your co-quarantiner has to look at his two cards and turn a deuce face up. It's an illusion to suppose this changes anything, right? It's still a bad game for you. But if there were yet another rule, one requiring you and the co-quarantiner to now swap the remaining downturned cards before looking to see who has the ace, the game would become a money-maker for you. If you're still not persuaded, just play a bunch of games and see who wins how often under what rules.

What I find intriguing about Smith's argument concerns what, if true, it suggests about our minds—namely, that they seem to be repulsed by randomness, prefer order, and therefore are constantly exercising themselves to discover patterns. It doesn't mean there aren't patterns. It means that detecting a pattern when there is none is a common mistake compared to failing to detect a pattern. I think this would explain a lot, like the hold of elaborate conspiracy theories on the human mind. Oswald was not enough of a marksman to have shot Kennedy! There must have been additional gunmen in Dealey Plaza and a conspiracy involving Lyndon Johnson, J. Edgar Hoover, and the Soviet Union! But what if Oswald just got lucky? (It happens, I once hit two doubles in the same baseball game.) A basketball player misses a foul shot that would have won the game. She choked! But wait, she doesn't make them all when practicing in her driveway. What if the fateful, errant shot was just one of those that she happens to miss? We're told that everything happens for a reason. I'll subscribe, if "random chance that might not have been realized but was" is allowed as a reason.

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