Comment author:XiXiDu
04 June 2010 09:55:03AM
1 point
[-]

Why would you need more than plain English to intuitively grasp Monty-Hall-type problems?

Take the original Monty Hall 'Dilemma'. Just imagine there are two candidates, A and B. A and B both choose the same door. After the moderator picked one door A always stays with his first choice, B always changes his choice to the remaining third door. Now imagine you run this experiment 999 times. What will happen? Because A always stays with his initial choice, he will win 333 cars. But where are the remaining 666 cars? Of course B won them!

Or conduct the experiment with 100 doors. Now let’s say the candidate picks door 8. By rule of the game the moderator now has to open 98 of the remaining 99 doors behind which there is no car. Afterwards there is only one door left besides door 8 that the candidate has chosen. Obviously you would change your decision now! The same should be the case with only 3 doors!

There really is no problem here. You don’t need to simulate this. Your chance of picking the car first time is 1/3 but your chance of choosing a door with a goat behind it, at the beginning, is 2/3. Thus on average, 2/3 of times that you are playing this game you’ll pick a goat at first go. That also means that 2/3 of times that you are playing this game, and by definition pick a goat, the moderator will have to pick the only remaining goat. Because given the laws of the game the moderator knows where the car is and is only allowed to open a door with a goat in it. What does that mean? That on average, at first go, you pick a goat 2/3 of the time and hence the moderator is forced to pick the remaining goat 2/3 of the time. That means 2/3 of the time there is no goat left, only the car is left behind the remaining door. Therefore 2/3 of the time the remaining door has the car.

I don't need fancy visuals or even formulas for this. Do you really?

Comment author:Kaj_Sotala
04 June 2010 04:04:51PM
10 points
[-]

I can testify that this isn't anywhere near as obvious to most people than it is to you. I, for one, had to have other people explain it to me the first time I ran into the problem, and even then it took a small while.

Comment author:XiXiDu
05 June 2010 07:00:46PM
1 point
[-]

I think the very problem in understanding such issues is shown in your reply. People assume too much, they read too much into things. I never said it has been obvious to me. I asked why you would need more than plain English to understand it and gave some examples on how to describe the problem in an abstract way that might be ample to grasp the problem sufficiently. If you take things more literally and don't come up with options that were never mentioned it would be much easier to understand. Like calling the police in the case of the trolley problem or whatever was never intended to be a rule of a particular game.

Comment author:Kaj_Sotala
06 June 2010 06:50:54AM
2 points
[-]

Well, yeah. But if I recall, I did have a plain English explanation of it. There was an article on Wikipedia about it, though since this was at least five years ago, the explanation wasn't as good as it is in today's article. It still did a passing job, though, which wasn't enough for me to get it very quickly.

Comment author:XiXiDu
06 June 2010 09:56:37AM
*
2 points
[-]

Yesterday, when falling asleep, I remembered that I indeed used the word 'obvious' in what I wrote. Forgot about it, I wrote the plain-English explanation from above earlier in a comment to the article 'Pigeons outperform humans at the Monty Hall Dilemma' and just copied it from there.

Anyway, I doubt it is obvious to anyone the first time. At least anyone who isn't a trained Bayesian. But for me it was enough to read some plain-English (German actually) explanations about it to come to the conclusion that the right solution is obviously right and now also intuitively so.

Maybe the problem is also that most people are simply skeptical to accept a given result. That is, is it really obvious to me now or have I just accepted that it is the right solution, repeated many times to become intuitively fixed? Is 1 + 1 = 2 really obvious? The last page of Russel and Whitehead's proof that 1+1=2 could be found on page 378 of the Principia Mathematica. So is it really obvious or have we simply all, collectively, come to accept this 'axiom' to be right and true?

I haven't had much time lately to get much further with my studies, I'm still struggling with basic Algebra. I have almost no formal education and try to educate myself now. That said, I started to watch a video series lately (The Most IMPORTANT Video You'll Ever See) and was struck when he said that to roughly figure out the doubling time you simply divide 70 by the percentage growth rate. I went to check it myself if it works and later looked it up. Well, it's NOT obvious why this is the case, at least not for me. Not even now that I have read up on the mathematical strict formula. But I'm sure, as I will think about it more, read more proofs and work with it, I'll come to regard it as obviously right. But will it be any more obvious than before? I will simply have collected some evidence for its truth value and its consistency. Things just start to make sense, or we think so because they work and/or are consistent.

Comment author:Blueberry
06 June 2010 05:44:54PM
2 points
[-]

But I'm sure, as I will think about it more, read more proofs and work with it, I'll come to regard it as obviously right. But will it be any more obvious than before?

If you're interested, here is a good explanation of the derivation of the formula. I don't think it's obvious, any more than the quadratic formula is obvious: it's just one of those mathematical tricks that you learn and becomes second nature.

Comment author:JoshuaZ
06 June 2010 05:53:45PM
*
0 points
[-]

I'm not sure I'm completely happy with that explanation. They use the result that ln(1+x) is very close to x when x is small. This is due to the Taylor series expansion of ln(1+x) (edit:or simply on looking at the ratio of the two and using L'Hospital's rule), but if one hasn't had calculus, that claim is going to look like magic.

Those explanations are really great. I've missed such in school when wondering WHY things behave like they do, when I was only shown HOW to use things to get what I want to do. But what do these explanations really explain. I think they are merely satisfying our idea that there is more to it than meets the eye. We think something is missing. What such explanations really do is to show us that the heuristics really work and that they are consistent on more than one level, they are reasonable.

Comment author:RobinZ
07 June 2010 04:03:25PM
*
1 point
[-]

That said, I started to watch a video series lately [...] and was struck when he said that to roughly figure out the doubling time you simply divide 70 by the percentage growth rate. I went to check it myself if it works and later looked it up. Well, it's NOT obvious why this is the case, at least not for me. Not even now that I have read up on the mathematical strict formula.

Well, it's an approximation, that's all. Pi is approximately equal to 355/113 - yeah, there's good mathematical reasons for choosing that particular fraction as an approximation, but the accuracy justifies itself. [edited sentence:] You only need one real revelation to not worry about how true Td = 70/r is: that the doubling time is a smooth line - there's no jaggedy peaks randomly in the middle. After that, you can just look how good the fit is and say, "yeah, that works for 0.1 < r < 20 for the accuracy I need".

Comment author:Martin-2
14 February 2013 04:32:33AM
*
0 points
[-]

Although it's late, I'd like to say that XiXiDu's approach deserves more credit and I think it would have helped me back when I didn't understand this problem. Eliezer's Bayes' Theorem post cites the percentage of doctors who get the breast cancer problem right when it's presented in different but mathematically equivalent forms. The doctors (and I) had an easier time when the problem was presented with quantities (100 out of 10,000 women) than with explicit probabilities (1% of women).

Likewise, thinking about a large number of trials can make the notion of probability easier to visualize in the Monty Hall problem. That's because running those trials and counting your winnings looks like something. The percent chance of winning once does not look like anything. Introducing the competitor was also a great touch since now the cars I don't win are easy to visualize too; that smug bastard has them!

Or you know what? Maybe none of that visualization stuff mattered. Maybe the key sentence is "[Candidate] A always stays with his first choice". If you commit to a certain door then you might as well wear a blindfold from that point forward. Then Monty can open all 3 doors if he likes and it won't bring your chances any closer to 1/2.

Comment author:logical
04 June 2010 04:43:08PM
-8 points
[-]

Are you serious? Are you buying this? Ok - let me make this easy: There NEVER WAS a 33% chance. Ever. The 1-in-3 choice is a ruse. No matter what door you choose, Monty has at least one door with a goat behind it, and he opens it. At that point, you are presented with a 1-in-2 choice. The prior choice is completely irrelevant at this point! You have a 50% chance of being right, just as you would expect. Your first choice did absolutely nothing to influence the outcome! This argument reminds me of the time I bet $100 on black at a roulette table because it had come up red for like 20 consecutive times, and of course it came up red again and I lost my $$. A guy at the table said to me "you really think the little ball remembers what it previously did and avoids the red slots??". Don't focus on the first choice, just look at the second - there's two doors and you have to choose one (the one you already picked, or the other one). You got a 50% chance.

Comment author:JoshuaZ
04 June 2010 04:57:19PM
*
10 points
[-]

Think about it this way. Let's say you precommit before we play Monty's game that you won't switch. Then you win 1/3rd of the time, exactly when you picked the correct door first, yes?

Now, suppose you precommit to switching. Under what circumstances will you win? You'll win if you didn't pick the correct door to start with. That means you have a 2/3rd chance of winning since you win whenever your first door wasn't the correct choice.

Your comparison to the roulette wheel doesn't work: The roulette wheel has no memory, but in this case, the car isn't reallocated between the two remaining doors, it was chosen before the process started.

Comment author:Sideways
04 June 2010 05:02:04PM
3 points
[-]

Your analogy doesn't hold, because each spin of the roulette wheel is a separate trial, while choosing a door and then having the option to choose another are causally linked.

If you've really thought about XiXiDu's analogies and they haven't helped, here's another; this is the one that made it obvious to me.

Omega transmutes a single grain of sand in a sandbag into a diamond, then pours the sand equally into three buckets. You choose one bucket for yourself. Omega then pours the sand from one of his two buckets into the other one, throws away the empty bucket, and offers to let you trade buckets.

Each bucket analogizes to a door that you may choose; the sand analogizes to probability mass. Seen this way, it's clear that what you want is to get as much sand (probability mass) as possible, and Omega's bucket has more sand in it. Monty's unopened door doesn't inherit anything tangible from the opened door, but it does inherit the opened door's probability mass.

Comment author:JoshuaZ
04 June 2010 05:08:36PM
2 points
[-]

That works better for you? That's deeply surprising. Using entities like Omega and transmutation seems to make things more abstract and much harder to understand what the heck is going on. I must need to massively update my notions about what sort of descriptors can make things clear to people.

Comment author:Sideways
04 June 2010 05:19:15PM
1 point
[-]

I use entities outside human experience in thought experiments for the sake of preventing Clever Humans from trying to game the analogy with their inferences.

"If Monty 'replaced' a grain of sand with a diamond then the diamond might be near the top, so I choose the first bucket."

"Monty wants to keep the diamond for himself, so if he's offering to trade with me, he probably thinks I have it and wants to get it back."

It might seem paradoxical, but using 'transmute at random' instead of 'replace', or 'Omega' instead of 'Monty Hall', actually simplifies the problem for me by establishing that all relevant facts to the problem have already been included. That never seems to happen in the real world, so the world of the analogy is usefully unreal.

Comment author:AlephNeil
05 June 2010 01:06:38AM
0 points
[-]

I'm not keen on this analogy because you're comparing the effect of the new information to an agent freely choosing to pour sand in a particular way. A confused person won't understand why Omega couldn't decide to distribute sand some other way - e.g. equally between the two remaining buckets.

Anyway, I think JoshuaZ's explanation is the clearest I've ever seen.

Comment author:logical
04 June 2010 05:13:11PM
-4 points
[-]

"Your analogy doesn't hold, because each spin of the roulette wheel is a separate trial, while choosing a door and then having the option to choose another are causally linked."

No, they are not causally linked. It does not matter what door you choose, you don't influence the outcome in any way at all. Ultimately, you have to choose between two doors. In fact, you don't "choose" a door at first at all. Because there is always at least one goat behind a door you didn't choose, you cannot influence the next action, which is for Monty to open a door with a goat. At that point it's a choice between two doors.

Comment author:JoshuaZ
04 June 2010 05:16:05PM
*
6 points
[-]

At this point you've had this explained to you multiple times. May I suggest that if you don't get it at this point, maybe be a bit of an empiricist and write a computer program to repeat the game many times and see what fraction switching wins? Or if you don't have the skill to do that (in which case learning to program should be on your list of things to learn how to do. It is very helpful and forces certain forms of careful thinking) play the game out with a friend in real life.

Comment author:Sideways
04 June 2010 05:51:05PM
5 points
[-]

If--and I mean do mean if, I wouldn't want to spoil the empirical test--logical doesn't understand the situation well enough to predict the correct outcome, there's a good chance he won't be able to program it into a computer correctly regardless of his programming skill. He'll program the computer to perform his misinterpretation of the problem, and it will return the result he expects.

On the other hand, if he's right about the Monty Hall problem and he programs it correctly... it will still return the result he expects.

Comment author:Sideways
04 June 2010 06:24:24PM
*
4 points
[-]

Sure, but then the question becomes whether the other programmer got the program right...

My point is that if you don't understand a situation, you can't reliably write a good computer simulation of it. So if logical believes that (to use your first link) James Tauber is wrong about the Monty Hall problem, he has no reason to believe Tauber can program a good simulation of it. And even if he can read Python code, and has no problem with Tauber's implementation, logical might well conclude that there was just some glitch in the code that he didn't notice--which happens to programmers regrettably often.

I think implementing the game with a friend is the better option here, for ease of implementation and strength of evidence. That's all :)

Comment author:TylerK
04 June 2010 06:22:16PM
5 points
[-]

The thing you might be overlooking is that Monty does not open a door at random, he opens a door guaranteed to contain a goat. When I first heard this problem, I didn't get it until that was explicitly pointed out to me.

If Monty opens a door at random (and the door could contain a car), then there is no causal link and therefore the probability would be as you describe.

## Comments (191)

BestWhy would you need more than plain English to intuitively grasp Monty-Hall-type problems?

Take the original Monty Hall 'Dilemma'. Just imagine there are two candidates, A and B. A and B both choose the same door. After the moderator picked one door A always stays with his first choice, B always changes his choice to the remaining third door. Now imagine you run this experiment 999 times. What will happen? Because A always stays with his initial choice, he will win 333 cars. But where are the remaining 666 cars? Of course B won them!

Or conduct the experiment with 100 doors. Now let’s say the candidate picks door 8. By rule of the game the moderator now has to open 98 of the remaining 99 doors behind which there is no car. Afterwards there is only one door left besides door 8 that the candidate has chosen. Obviously you would change your decision now! The same should be the case with only 3 doors!

There really is no problem here. You don’t need to simulate this. Your chance of picking the car first time is 1/3 but your chance of choosing a door with a goat behind it, at the beginning, is 2/3. Thus on average, 2/3 of times that you are playing this game you’ll pick a goat at first go. That also means that 2/3 of times that you are playing this game, and by definition pick a goat, the moderator will have to pick the only remaining goat. Because given the laws of the game the moderator knows where the car is and is only allowed to open a door with a goat in it. What does that mean? That on average, at first go, you pick a goat 2/3 of the time and hence the moderator is forced to pick the remaining goat 2/3 of the time. That means 2/3 of the time there is no goat left, only the car is left behind the remaining door. Therefore 2/3 of the time the remaining door has the car.

I don't need fancy visuals or even formulas for this. Do you really?

I can testify that this isn't anywhere near as obvious to most people than it is to you. I, for one, had to have other people explain it to me the first time I ran into the problem, and even then it took a small while.

I think the very problem in understanding such issues is shown in your reply. People assume too much, they read too much into things. I never said it has been obvious to me. I asked why you would need more than plain English to understand it and gave some examples on how to describe the problem in an abstract way that might be ample to grasp the problem sufficiently. If you take things more literally and don't come up with options that were never mentioned it would be much easier to understand. Like calling the police in the case of the trolley problem or whatever was never intended to be a rule of a particular game.

Well, yeah. But if I recall, I

didhave a plain English explanation of it. There was an article on Wikipedia about it, though since this was at least five years ago, the explanation wasn't as good as it is in today's article. It still did a passing job, though, which wasn't enough for me to get it very quickly.*2 points [-]Yesterday, when falling asleep, I remembered that I indeed used the word 'obvious' in what I wrote. Forgot about it, I wrote the plain-English explanation from above earlier in a comment to the article 'Pigeons outperform humans at the Monty Hall Dilemma' and just copied it from there.

Anyway, I doubt it is obvious to anyone the first time. At least anyone who isn't a trained Bayesian. But for me it was enough to read some plain-English (German actually) explanations about it to come to the conclusion that the right solution is obviously right and now also intuitively so.

Maybe the problem is also that most people are simply skeptical to accept a given result. That is, is it really

obviousto me now or have I just accepted that it is the right solution, repeated many times to become intuitively fixed? Is 1 + 1 = 2 really obvious? The last page of Russel and Whitehead's proof that 1+1=2 could be found on page 378 of the Principia Mathematica. So is it really obvious or have we simply all, collectively, come to accept this 'axiom' to berightandtrue?I haven't had much time lately to get much further with my studies, I'm still struggling with basic Algebra. I have almost no formal education and try to educate myself now. That said, I started to watch a video series lately (The Most IMPORTANT Video You'll Ever See) and was struck when he said that to roughly figure out the doubling time you simply divide 70 by the percentage growth rate. I went to check it myself if it works and later looked it up. Well, it's NOT obvious why this is the case, at least not for me. Not even now that I have read up on the mathematical strict formula. But I'm sure, as I will think about it more, read more proofs and work with it, I'll come to regard it as obviously right. But will it be any more obvious than before? I will simply have collected some evidence for its truth value and its consistency. Things just start to make sense, or we think so because they work and/or are consistent.

If you're interested, here is a good explanation of the derivation of the formula. I don't think it's obvious, any more than the quadratic formula is obvious: it's just one of those mathematical tricks that you learn and becomes second nature.

*0 points [-]I'm not sure I'm completely happy with that explanation. They use the result that ln(1+x) is very close to x when x is small. This is due to the Taylor series expansion of ln(1+x) (edit:or simply on looking at the ratio of the two and using L'Hospital's rule), but if one hasn't had calculus, that claim is going to look like magic.

*2 points [-]Here are more examples:

Those explanations are really great. I've missed such in school when wondering WHY things behave like they do, when I was only shown HOW to use things to get what I want to do. But what do these explanations really

explain. I think they are merely satisfying our idea that there ismore to itthan meets the eye. We think something is missing. What such explanations really do is to show us that the heuristics really work and that they are consistent on more than one level, they arereasonable.*1 point [-]Well, it's an approximation, that's all. Pi is approximately equal to 355/113 - yeah, there's good mathematical reasons for choosing that particular fraction as an approximation, but the accuracy justifies itself.

[edited sentence:]You only need one realrevelationtonot worry abouthow trueTd= 70/ris: that the doubling time is a smooth line - there's no jaggedy peaks randomly in the middle. After that, you can just look how good the fit is and say, "yeah, that works for 0.1 <r< 20 for the accuracy I need".*0 points [-]Although it's late, I'd like to say that XiXiDu's approach deserves more credit and I think it would have helped me back when I didn't understand this problem. Eliezer's Bayes' Theorem post cites the percentage of doctors who get the breast cancer problem right when it's presented in different but mathematically equivalent forms. The doctors (and I) had an easier time when the problem was presented with quantities (100 out of 10,000 women) than with explicit probabilities (1% of women).

Likewise, thinking about a large number of trials can make the notion of probability easier to visualize in the Monty Hall problem. That's because running those trials and counting your winnings

lookslike something. The percent chance of winning once does not look like anything. Introducing the competitor was also a great touch since now the cars I don't win are easy to visualize too; that smug bastard has them!Or you know what? Maybe none of that visualization stuff mattered. Maybe the key sentence is "[Candidate] A always stays with his first choice". If you commit to a certain door then you might as well wear a blindfold from that point forward. Then Monty can open all 3 doors if he likes and it won't bring your chances any closer to 1/2.