Two New Ideas For Today
Jan. 9th, 2010 06:31 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
snousleI've been on a real reading binge lately.
I am amazed that after so many decades, the Monty Hall Problem still yields fresh insights.
An unsigned editorial from the Economist discussed it recently, and concludes with a thought that had never occurred to me: why would anyone assume that the game show host wants to give you a car?
This comment is relevant to the version of the game where the conventional rules are not known to the contestant, who has never seen the game before. Curiously, the "intuitive" answer, which is incorrect under the "standard" formulation of the game, is arguably "not totally wrong" when the contestant intuitively assumes that the host is trying to keep the car for himself, or that the host is likely to cheat.
Now that I think of it, I realize that many years ago, on first being presented with the ignorant-contestant version of the problem, my assumption that the show host would give me information, by revealing the goat according to a set of rather generous rules, was just as unwarranted as the assumption that he would be trying to cheat me.
People who attempt to solve the problem, and who use the correct mathematical reasoning in the process, for some reason almost always seem to assume the standard rules without prompting. But when you think of it from the host's perspective, the standard rules don't make a whole lot of sense. I have never seen "Monty From Hell" assumed as a basis for correct reasoning, although there is no reason not to. Why is this? Maybe it's because the people who best understand probability are all namby-pamby liberal academics who project the "nurturing mother" archetype onto the game show host, using the same framing that Lakoff attributes to left-wing politics? Must be! It's just another part of the vast liberal conspiracy.
[Or, perhaps the game is a conspiracy on the part of auto manufacturers; maybe to them, it's not about probability at all, but rather an effort to reenforce a value system that favors their interests while distracting their customers with a shiny mathematical trinket. After all, it hardly ever occurs to anyone that a goat might be more desirable than a car.]
Anyway, a warning: discussing Monty Hall with your significant other substantially raises the probability of divorce.
If you are wondering how anyone could spend so much time on such a simple problem, here's an explanation: I just discovered that I'm a Bayesian Initiate. This is way more sinister than black helicopters, the Trilateral Commission, the New World Order, or the Freemasons, because it's all right there in plain sight. We rule the world for one simple reason: we're so fucking boring, nobody gives a damn what we do. LOL.
The article is only half kidding. Bayes' Theorem rewired my brain long before you ever had a spam filter. Monty Hall is like a station of the cross for us.
The other obsession du jour: Newton's Cradle. I slept badly last night, so in the wee hours I was thinking about physics. I had a sudden insight from something long, long ago - the high-school treatment of this problem admits more solutions than they tell you about, and the theory they present you with gives you a sense that you "understand" the system when in fact you have only scratched the surface. Conservation of energy and momentum, in a simple calculation, does predict the outcome you see in reality, but there's a hidden assumption in there that selects one particular outcome out of an infinity of solutions - one that happens to match what you see. What happens is, they show you the device, run through the calculations, and don't mention that they have sneakily embedded the key to the already-known outcome in the assumptions. I bet if you had to solve the problem from scratch, having never seen the device, you wouldn't derive the outcome correctly. Because given the theory they teach you, you can't. You can only show, retrospectively, that conservation of energy is not violated.
I figured I must have missed something here, so I hit the Web. Hot damn, I was right! Finding a fuller account was kind of shocking, because if you'd asked me yesterday, I would have told you that I already knew all there was to know about this. Goes to show you...
Now, I'm not saying this is a conspiracy or anything - this device occupies, maybe, fifteen minutes of class time, and having run briskly through the simple account, you move on to the next thing. In truth, the full account is messy, the simple account is nice, and you really do learn something useful from it. Plus, it's shiny and makes a curiously satisfying clicking noise. So I don't actually advocate changing the curriculum. But dang, is it ever a good example of how problems can get declared prematurely solved. It's almost better as a philosophy-of-science problem than a physics problem.
I am amazed that after so many decades, the Monty Hall Problem still yields fresh insights.
An unsigned editorial from the Economist discussed it recently, and concludes with a thought that had never occurred to me: why would anyone assume that the game show host wants to give you a car?
This comment is relevant to the version of the game where the conventional rules are not known to the contestant, who has never seen the game before. Curiously, the "intuitive" answer, which is incorrect under the "standard" formulation of the game, is arguably "not totally wrong" when the contestant intuitively assumes that the host is trying to keep the car for himself, or that the host is likely to cheat.
Now that I think of it, I realize that many years ago, on first being presented with the ignorant-contestant version of the problem, my assumption that the show host would give me information, by revealing the goat according to a set of rather generous rules, was just as unwarranted as the assumption that he would be trying to cheat me.
People who attempt to solve the problem, and who use the correct mathematical reasoning in the process, for some reason almost always seem to assume the standard rules without prompting. But when you think of it from the host's perspective, the standard rules don't make a whole lot of sense. I have never seen "Monty From Hell" assumed as a basis for correct reasoning, although there is no reason not to. Why is this? Maybe it's because the people who best understand probability are all namby-pamby liberal academics who project the "nurturing mother" archetype onto the game show host, using the same framing that Lakoff attributes to left-wing politics? Must be! It's just another part of the vast liberal conspiracy.
[Or, perhaps the game is a conspiracy on the part of auto manufacturers; maybe to them, it's not about probability at all, but rather an effort to reenforce a value system that favors their interests while distracting their customers with a shiny mathematical trinket. After all, it hardly ever occurs to anyone that a goat might be more desirable than a car.]
Anyway, a warning: discussing Monty Hall with your significant other substantially raises the probability of divorce.
If you are wondering how anyone could spend so much time on such a simple problem, here's an explanation: I just discovered that I'm a Bayesian Initiate. This is way more sinister than black helicopters, the Trilateral Commission, the New World Order, or the Freemasons, because it's all right there in plain sight. We rule the world for one simple reason: we're so fucking boring, nobody gives a damn what we do. LOL.
The article is only half kidding. Bayes' Theorem rewired my brain long before you ever had a spam filter. Monty Hall is like a station of the cross for us.
The other obsession du jour: Newton's Cradle. I slept badly last night, so in the wee hours I was thinking about physics. I had a sudden insight from something long, long ago - the high-school treatment of this problem admits more solutions than they tell you about, and the theory they present you with gives you a sense that you "understand" the system when in fact you have only scratched the surface. Conservation of energy and momentum, in a simple calculation, does predict the outcome you see in reality, but there's a hidden assumption in there that selects one particular outcome out of an infinity of solutions - one that happens to match what you see. What happens is, they show you the device, run through the calculations, and don't mention that they have sneakily embedded the key to the already-known outcome in the assumptions. I bet if you had to solve the problem from scratch, having never seen the device, you wouldn't derive the outcome correctly. Because given the theory they teach you, you can't. You can only show, retrospectively, that conservation of energy is not violated.
I figured I must have missed something here, so I hit the Web. Hot damn, I was right! Finding a fuller account was kind of shocking, because if you'd asked me yesterday, I would have told you that I already knew all there was to know about this. Goes to show you...
Now, I'm not saying this is a conspiracy or anything - this device occupies, maybe, fifteen minutes of class time, and having run briskly through the simple account, you move on to the next thing. In truth, the full account is messy, the simple account is nice, and you really do learn something useful from it. Plus, it's shiny and makes a curiously satisfying clicking noise. So I don't actually advocate changing the curriculum. But dang, is it ever a good example of how problems can get declared prematurely solved. It's almost better as a philosophy-of-science problem than a physics problem.
