One Goat Short

I like game shows. Liking game shows is not one of the more respectable hobbies, compared to, say, Crimean War pedantry, or laughing at goats. Game shows have a long history of being sneered at by people who can’t be bothered to learn enough about game shows to sneer at them for correct reasons. Lost somewhere within my archives is even an anthology of science fiction short stories about game shows, which if you take out the punch lines of “and the loser DIES!” or “and the host [ typically Chuck Woolery ] is SATAN!”, would leave nearly nothing, and considering that science fiction as a genre has spent most of its existence feeling picked-on as the “smelly, unemployed cousin of the entertainment industry” (Mike Nelson’s Movie Megacheese) that’s quite some sneering. Sneering at game shows even earned an episode of The Mary Tyler Moore show which managed to be not just bad but offensively illogical.

Nevertheless, I like them, and was a child at a great age for game shows on broadcast television: the late 1970s and early 1980s had an apparently endless menu of programs, testing people’s abilities to think of words, to spell words, to price household goods, and guess how other people answered surveys. We haven’t anything like that anymore; on network TV about the only game shows that survive are Jeopardy! (which nearly alone of the genre gets any respect), Wheel of Fortune, The Price Is Right, and, returned after decades away, Let’s Make A Deal. (I don’t regard reality shows as game shows, despite a common programming heritage. I can’t say what it is precisely other than location and sometimes scale that, say, Survivor or The Amazing Race do that Beat The Clock or Truth Or Consequences do not, but there’s something.) Now and then something new flutters into being, but it vanishes without leaving much of a trace, besides retreading jokes about the people who’d watch it.

All that is longwinded preliminary to one of those things that amuses mostly me. On the Thursday (27 October) episode of Let’s Make A Deal, they briefly looked like they might be playing The Monty Hall Problem.

The Monty Hall Problem is a reasonably famous puzzle, of the kind really good for recreational mathematics in that it can be stated without any of the mathematics standing out. And since the right answer depends on understanding probability, most likely nobody who hears it will come up with the right answer, and for that matter, even if someone has the right answer it will probably be for the wrong reasons. Human beings deal with probability and the particularly troublesome subject of conditional probability every day in nearly every decision; unfortunately, we are all boundlessly rotten at probability problems. Certainly I got it wrong the first time I heard it; I even managed to prove it wrongly using a computer (a Commodore 128 with a faulty video chip, although I don’t believe the faulty video chip had anything to do with the problem).

The problem, as it’s generally put, is this: the contestant has free choice of what’s behind one of three curtains. Behind one is the truly desired prize, typically a car; behind the other two is a zonk, in the normal phrasing, a goat. After the contestant has made the choice, the host shows one of the zonk curtains, and offers a chance to switch. Is the contestant likely to do any better by switching?

I know you have an answer. Don’t share it. The correct answer depends on assumptions which have to be made explicit. For the most important, must the host show a curtain, or is it a free choice depending on how the show seems to be going and whether the contestant picked rightly to start? For the next most important, does the host know which curtain contains the car and which have the goats? You can’t say what the correct strategy is until you have fuller data, so, if you’re presented with a problem like this feel free to use the need to have the problem made more specifically correct to stall for time.

They didn’t actually play The Monty Hall Problem, of course. The contestant got her pick (number three) of three curtains, and number two (a vacation package) was revealed. She was then assured there was one zonk left; did she want to stick with her choice, or — no, not swap, but, to take a payoff of a thousand dollars to give up the curtain. She chose the money, understandably, although it turns out she had picked the curtain with a car behind it. The zonk was a Shoe Car.

But there’s an amusing bit of mathematics lore behind all this anyway. Let’s Make A Deal has never, as best as anyone’s found, played The Monty Hall Problem. It’s played games from which one can see The Monty Hall Problem, particularly in its steady stream of asking contestants to second-guess themselves and to reveal partial information as a way of making the second-guessing harder. But it’s never actually done the game named for it.

This isn’t all that startling. To a first approximation, nothing in mathematics is named for whatever it ought to be named for. My favorite example of this misattribution is the Pell Equation, the problem of finding pairs of whole numbers, x and y, so that the square of x is equal to one plus some whole multiple of the square of y. It’s named for John Pell, a 17th century English mathematician, and also Oliver Cromwell’s political agent to Protestant Switzerland. But Pell didn’t discover the equation; forms of it have been known and used as much as three thousand years ago. He didn’t put it in its modern and nicely general form; Pierre de Fermat (that Fermat) did. He didn’t solve it. His entire connection to it appears to have been that he edited an algebra book (written by Johann Rahn, who I don’t know anything interesting about), which contained solutions of problems posed by John Wallis (among other things, the person who introduced the symbol ∞ to the world) and William Brouncker (first president of the Royal Society, and a Commissioner of the Navy with whom Samuel Pepys quarreled), including this famous one. Leonhard Euler apparently conflated Brouncker, who worked on the problem, with Pell, gave the name Pell’s Equation to it, and it stuck.

In comparison, the name of The Monty Hall Problem is not too bad. Let’s Make A Deal does after all have goats and cars.

(I admit I do miss the long-since disregarded custom of swapping a part of the contestant’s outfit for the deal. It may have been a formality and only noticed in the first years of Monty Hall’s run on the show, but the name of the game does after all imply deal-making, and swapping a button for a car may be a lopsided deal but it is a deal. Swapping knowledge of in which decade Cap’n Crunch was first marketed for a car is not as satisfyingly tangible, and it tips off the audience when some part of the costume is demanded that there’s going to be a zonk featuring contestants’ hats or buttons or the like.)