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.
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no subject
Date: 2010-01-10 04:40 am (UTC)no subject
Date: 2010-01-10 05:04 am (UTC)2) Why stick?
no subject
Date: 2010-01-10 05:31 am (UTC)"Suppose you're on a game show and you're given the choice of three doors. Behind one door is a car; behind the others, goats. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one randomly. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you "Do you want to switch to Door Number 2?" Is it to your advantage to change your choice?"
In considering the rules for the host, is this the version in which you say you should stick with your original choice?
If you check the Wikipedia page it lists all the variants, and the win probability for the stick/switch scenario is different in each case.
I would be very interested to know if you think the account on the Wikipedia page is incorrect in any way.
no subject
Date: 2010-01-10 05:47 am (UTC)I like this approach to the problem because it cuts through all the butt-extracted chatroom analysis people proffer and forces the problem into the empirical regime. It's hard to maintain that both strategies are equally effective when you've just helped demonstrate they're not.
And then your spouse hits you (or vice versa) with the most important question: "Okay, the strategies are not equal. Why aren't they equal?" Then, and only then, is it time to start doing math.
*sigh* So many problems could be solved with a small dose of empiricism.
[I got the breast cancer screening analysis wrong. :( But I was in the ballpark, and recognized that the test was ineffective. I need to do more work to get my Bayesian Initiate merit badge.]
no subject
Date: 2010-01-10 06:00 am (UTC)So what did I learn? Some people are unwilling to test their beliefs against reality, even in the most trivially simple case. If people fight tooth and nail against admitting the possibility that they could be wrong, well... what does say about our chance for reasoned debate of issues that actually matter?
no subject
Date: 2010-01-10 06:01 am (UTC)made me successfulearned me a lot of money in a science career was being a natural Bayesian. Monty Hall has always been crystal clear. The most significant work I did - and now that I think of it, most of the insignificant stuff too - was all founded on this style of reasoning.I'm having an I [heart] Wikipedia day today.
no subject
Date: 2010-01-10 06:06 am (UTC)I realize I've never played the actual game. I mean, I've done a zillion computer simulations that empirically validate things of very similar structure, but I've never done it with actual cars and goats (or reasonable facsimiles thereof).
We should do that sometime! (possible spring visit?)
no subject
Date: 2010-01-10 06:58 am (UTC)I can just picture it.
Date: 2010-01-10 06:59 am (UTC)"Snowboard. And try to win as few goats as I can."
no subject
Date: 2010-01-10 07:11 am (UTC)Hope you like goats though. ;-)
no subject
Date: 2010-01-10 07:22 am (UTC)Game show with winning cars = Better ratings and host gets a second season.
Therefore, game show host wants cars to be given away.
no subject
Date: 2010-01-10 09:41 am (UTC)I sold my tolerance for ambiguity for beer money, can you tell?
no subject
Date: 2010-01-10 05:12 pm (UTC)no subject
Date: 2010-01-10 05:40 pm (UTC)There is all kinds of interesting research going on concerning how to explain probabilistic reasoning to those who are unfamiliar with it. Very important stuff, especially with things like the cancer-diagnosis problem. I really like this, because many
wankerswriters are satisfied with pounding on the "people are so stupid" theme... this goes beyond that into something constructive.no subject
Date: 2010-01-11 06:27 am (UTC)It's very hard for people to develop intuition for many probability problem types.
Meanwhile, I hadn't read Martin Gardner's name in a long, long time now. From the Wiki article, I found out he published a lot more than I knew. Good 'ol recreational mathematics.
no subject
Date: 2010-01-11 06:38 am (UTC){g}
no subject
Date: 2010-01-14 07:58 pm (UTC)