The Monty Hall problem

[Rolfe]

That's the definition of "The Monty Hall Problem."

Perhaps the OP did not state all of the details of the problem, assuming that the reader would know them all, but if you look up the problem on Wikipedia, they're there. http://en.wikipedia.org/wiki/Monty_Hall_problem .

How Monty actually behaved in his actual show isn't really relevant at this point - "The Monty Hall Problem" is a well-defined problem with a counterintuitive but well-defined solution.

Frankly, I'm not interested in what Wikipedia has to say, since I first solved this problem long before Wikipedia was born or thought of, and if I felt like it I could go edit the entry right now to read differently.

I have never seen anyone in the position of the OP come up with a fully-defined version of the puzzle. Invariably, the discussion progresses to the assumptions one has to make in order to solve the problem in the way it was probably originally intended to be solved. Only if it is clear that Monty is obliged to open a second door, and that when he does so he is obliged to reveal a goat, does the "well-defined" problem with its well-defined solution apply.

Simply relating what happens on a single iteration of the game, without any information about the ground rules or what might happen on other iterations, is not sufficient to solve the puzzle in the simple way in which it is usually understood.

Rolfe.

 

[JoelKatz]

I don't see why this is complicated, or why it matters whether or not Monty had a choice. What we know is that he opened a door with a goat, and that there will always be at least one door with a goat whatever was behind the first door you chose. The result would be the same if, instead of opening that door, Monty simply invited to you to swap your choice from the one door you first chose to all the doors you did not. Your chance of being right when you switch is still the same as your chance of being wrong on your first choice.

It's complicated because if Monty chose which door to open randomly, he won't be able choose a goat every time. So whatever the probability is in this case, it isn't the probability that you will win the game.

Consider: You pick a door. Monty shows a goat. You switch. Your door is opened. It reveals that you won. The odds that you won are 100%. However, since this will not happen every time, it doesn't mean that the odds that you will win in the situation as a whole are 100%.

Similarly, in the case where Monty does choose a goat, you will win 1/2 the time. However, that does not tell us the odds you will win in the original problem, which are 1/3.

Let's call the door you chose A and the door Monty opens B. There are three equally-likely possibilities, prize behind A, prize behind B, and prize behind C. Given that Monty shows you a goat, we know the prize cannot be behind B. That leaves two equally-likely cases, prize behind A, prize behind C. So whether you switch or not, your odds are fifty-fifty. (But this is only in the case where Monty shows you a goat which will not happen every time. So your overall odds of winning are not fifty-fifty!)

In the case where Monty chooses a door randomly and you switch, you will win 1/3 of the time. Call the door you pick A and the door Monty shows you B. You will wind up with door C. You will win if the prize is behind door C, which it will be 1/3 of the time. Only some of the time, of course, will a goat be behind door B. 1/3 of the time, Monty will show you the prize, and you'll be left to decide which losing door to open.

In the original problem, Monty chooses which door to open with knowledge of which one has the prize. Thus Monty is giving you information that you can take advantage of by switching.

A good real-world example of the 'Monty doesn't know' case is Deal or No Deal. In this game, you pick a suitcase. Other suitcases are opened randomly. At the end, say, there are two suitcases, one holding $1 and one holding $1,000,000. It should be obvious that either is equally likely to contain the $1,000,000. The only information you gained by opening suitcases that you chose is that only the $1 and $1,000,000 are left.

 

[humber]

You are quite wrong. If the host is clueless, then the goat tells you nothing except that particular door is wrong. But if the host knows and is constrained by your pick, then his choice carries more information. This is clear from the "pick two but say the third" equivalency if the host knows, which does not work if the host does not.

If a goat is revealed, then it makes no difference if that is done by intent or not. The added information provided by the goat, and suggesting swapping as the best option, does not come from Monty, but knowing that a car is there. .

You can also go through the cases or write a simulation if you wish. You'll see that both staying and switching win half the time that the host didn't accidentally reveal the car. I've done exactly that in a previous thread on this problem.

If Monty reveals the car, then the game is over...
It's the timing of the goat information that is interesting. If the goat is revealed prior to the contestant's choice, then the chance is 50/50, so there is no motive to swap, but if revealed after the contestant's choice, then it becomes advantageous to swap.
It's conditional probability, and it is the added information that makes it that way. One resorts to probability, when there is insufficient information.

ETA: That a car is there, is information that is common to Monty and the contestant. Monty also knows where the car is, but within the scope of the problem as stated by Vos Savant, that specific knowledge is of no use to him, unless he wants to show the car.

 

[Rasmus]

If Monty reveals the car, then the game is over...

Yes. But that will happen 1/3 of the time.

It's the timing of the goat information that is interesting. If the goat is revealed prior to the contestant's choice, then the chance is 50/50, so there is no motive to swap, but if revealed after the contestant's choice, then it becomes advantageous to swap.

1/3 -> you originally picked the car, Monty would always show you a goat and you will always lose if you switch.

2/3 -> you originally picked a goat. Now Monty has a 50% chance of showing you the car and ending the game prematurely (1/3 of all original cases!) and 50% of showing you the one remaining goat. (Another 1/3 of all original cases.) Here, you should obviously switch!

-> There are three total scenarios:

A) You pick the car and should not switch
B) You pick a goat and Monty shows you the car, thus ending the game and leaving you with a goat.
C) You pick a goat and Monty shows you the other goat. Here, you should switch.

It's 50/50.



It's conditional probability, and it is the added information that makes it that way. One resorts to probability, when there is insufficient information.

ETA: That a car is there, is information that is common to Monty and the contestant. Monty also knows where the car is, but within the scope of the problem as stated by Vos Savant, that specific knowledge is of no use to him, unless he wants to show the car.[/quote]

 

[humber]

Yes. But that will happen 1/3 of the time.

If he doesn't know, yes. But it's a game show, and the question says he does know.

1/3 -> you originally picked the car, Monty would always show you a goat and you will always lose if you switch.
2/3 -> you originally picked a goat. Now Monty has a 50% chance of showing you the car and ending the game prematurely (1/3 of all original cases!) and 50% of showing you the one remaining goat. (Another 1/3 of all original cases.) Here, you should obviously switch!
-> There are three total scenarios:
A) You pick the car and should not switch
B) You pick a goat and Monty shows you the car, thus ending the game and leaving you with a goat.
C) You pick a goat and Monty shows you the other goat. Here, you should switch.
It's 50/50.

Car shown at any time...game over. Probability = 1
Goat shown prior to choice 1/2, 1/2
Goat shown after choice, 1/3, 2/3
It's not "winning" but the probability of the car being revealed when a door is opened.

 

[Rolfe]

Things that need to be established before the ground rules are clear.

Can Monty choose whether or not he opens a second door at all? If he can, the problem is insoluble, because you don't know the basis for his choice. He may want to minimise the number of cars given out, and so only offer the choice if you already picked the car. Or he may think you have a nice face and so offer the choice when you picked wrong.

Can Monty choose whether or not he reveals the car when he opens the second door? If he can, then again there is insufficient information to answer. He may again be so keen to minimise the number of cars given out that he will always reveal the car if he can - thus meaning that switching is a 100% bad idea.

Neither of these makes it a valid intellectual puzzle, of course. It's very unlikely the poser of the question really means these to be taken into consideration. However, they are possible real-life scenarios that could well exist, and which come up for consideration as soon as the question rules/assumptions is raised.

The real Monty apparently varied his ground rules from example to example, thus keeping everyone guessing. He was capricious (literally), and so couldn't be second-guessed.

When turning the game into an intellectual puzzle, however, it is important to codify the rules so that capriciousness is elminated. So first, is has to be specified that Monty cannot alter his behaviour based on whether or not the car has already been chosen. So he must open a second door, and he mustn't deliberately choose the car to prevent it being won.

This still leaves the two scenarios under discussion - either Monty has to pick either of the two remaining doors at random (regardless of whether the car is then revealed), or he has to reveal a goat.

Even if you don't know which of these two scenarios is in operation, so long as you know it has to be one or the other, the answer is still "Oh God not again, switch". Because even in the former scenario, switching doesn't reduce your chance of winning, and in the latter, it doubles it. Switching may help, and it cannot hurt.

The thing that as always fascinated me about this is that whether or not switching will improve your chance of winning seems to depend on Monty's intentions when he opens that second door. Why should his intention matter a damn? It's really because the probability of any particular outcome can only be appreciated by considering the example in front of us as one of a set of iterations of the puzzle. With one intention, you get one set of iterations where he reveals the car a third of the time and it doesn't matter a damn if you switch, and with a different intention he never reveals the car, and you win twice as often if you switch. It's a question of which set does your single example belong to, and the thing that differentiates the two sets is Monty's intention.

Rolfe.

 

[sol invictus]

The thing that as always fascinated me about this is that whether or not switching will improve your chance of winning seems to depend on Monty's intentions when he opens that second door. Why should his intention matter a damn? It's really because the probability of any particular outcome can only be appreciated by considering the example in front of us as one of a set of iterations of the puzzle. With one intention, you get one set of iterations where he reveals the car a third of the time and it doesn't matter a damn if you switch, and with a different intention he never reveals the car, and you win twice as often if you switch. It's a question of which set does your single example belong to, and the thing that differentiates the two sets is Monty's intention.

Another way of looking at it is that messages contain useful information only when they are neither completely predictable nor completely random.

If Monty's possible actions are constrained by some set of rules known to you, then his actual actions convey information (so long as the rules aren't so restrictive that he has only one choice in all circumstances).

His intentions matter because they might determine or affect those rules - but what it really boils down to is what you know about those rules.

 

[humber]

The thing that as always fascinated me about this is that whether or not switching will improve your chance of winning seems to depend on Monty's intentions when he opens that second door.
Why should his intention matter a damn? It's really because the probability of any particular outcome can only be appreciated by considering the example in front of us as one of a set of iterations of the puzzle. With one intention, you get one set of iterations where he reveals the car a third of the time and it doesn't matter a damn if you switch, and with a different intention he never reveals the car, and you win twice as often if you switch. It's a question of which set does your single example belong to, and the thing that differentiates the two sets is Monty's intention.
Rolfe.

Yes. Otherwise it's a quotidian exercise.
The question of his presumed intent is rather settled by the desired outcome of not purposefully giving the car away. I see the interesting thing as being where the information is, and how that moves around with circumstance and sequence. His intent can be over-ruled by both. As far as I can see, and within the frame of Vos Savant's problem, he can only act intentionally to give the car away.

Vos Savant's version is not a rigorously defined, because it isn't an intellectual problem, it's to engender this sort of thing.

How Monty actually behaved in his actual show isn't really relevant at this point - "The Monty Hall Problem" is a well-defined problem with a counterintuitive but well-defined solution

Puzzles like this are often a sort of "knock-knock" joke. The "obvious" answers of 1/3 or 1/2 can be rejected on that basis, and 2/3 given as the remaining default answer, if nothing else.

I have a pet peeve about this, and the assertion of a "correct" answer. It's a form of intellectual snobbery, where there is another way of "knowing"; intuition, or the pure light of I.Q.

Vos Savant markets that very thing, and makes many errors in the process.
An apologist wrote this:

"Marilyn is not a civil engineer, electrical engineer, or ocean navigator. In these areas she is apt to blunder."

About this:
"Some answers may be so naive that one wonders if she is serious, like:

Q. When lost at sea, how do you find land?
A. Follow the waves, for waves always crash on a shore.

Oh, she's serious, because an IQ of 228 allows commonsense, let alone oceanography to be ignored.

The idea is to "see" that it's 2/3, and not work the math to show how contrived the problem is to achieve that magic.

 

[Rolfe]

Another way of looking at it is that messages contain useful information only when they are neither completely predictable nor completely random.

If Monty's possible actions are constrained by some set of rules known to you, then his actual actions convey information (so long as the rules aren't so restrictive that he has only one choice in all circumstances).

His intentions matter because they might determine or affect those rules - but what it really boils down to is what you know about those rules.

Well put. If the problem is formulated so as to convey that Monty must open a second door, and that this act must reveal a goat, it's pretty simple to show that switching doubles your chance of winning.

What tends to happen though is that the problem is not formulated so rigorously, and when this obvious trusim is explained, someone then says, "oh but you're assuming he has to show you a goat, that wasn't specified." And once you start down that road, the possibility that he might not have to open a second door at all (and may choose to do so with unknown motives), or could choose to reveal the car (and so prevent you having the opportunity to switch), then gets dragged in.

Which is all actually quite interseting, in that it widens the scope of the puzzle and leads to a set of varying answers depending on the set of assumptions you choose to infer. Some people, working on Game Theory, like to assume Evil Monty who will only offer you the switch if you have already chosen the car, and/or will reveal the car to prevent you winning it.

Still others like to imagine quantum wormholes that will swalow the prize, or homicidal accomplices that will shoot a winning contestant, but that's a bit left-field for normal mortals.

Rolfe.

 

[Rolfe]

The question of his presumed intent is rather settled by the desired outcome of not purposefully giving the car away.

But again, where is it said that we should presume this?

If he does want to avoid giving the car away, then obviously he only offers the switch if the contestant has already picked the car. Which then becomes counterproductive once enough iterations are in the public domain for this strategy to be deduced.

It seems fairly obvious that the problem as originally posed, is intended to cover the situation where Monty always opens a second door and always reveals a car, leading to the brain-teasing needed to understand that in that situation switching doubles your chances of winning. The problem lies in the puzzle not being rigorously enough defined.

I've seen better attempts where it's posed not as a human choosing which doors to open, but an unthinking mechanical system which will always open a second door and always reveal a goat. That way you can get to the "yes, switching will double your chances of winning" answer with a lot less angst.

I'm not at all familiar with Marilyn Vos savant in this context, though I've seen the name mentioned in this context. Did she go for unwarranted assumptions, or what? (As I understand the usual IQ scales, they max out somewhere in the 170s, so I don't think I believe in anyone with an alleged IQ of 228.)

Rolfe.

 

[Rasmus]

The question of his presumed intent is rather settled by the desired outcome of not purposefully giving the car away.

It doesn't say that anywhere, does it?

And I think it is far from obvious: The show works and brings in advertising money *because* a lot of prices are given away as well as a lot of prices being lost by people who had them or could have had them.

So Monty's intend would be to give away some cars, but not too many and not too often.

There used to be a similar show in Germany. Many years later the host was on a different show, demonstrating how he could easily steer people to do what he wanted. I seems to remember - from the original show - that throughout the game one contestant gave up the car no less than three times when he swapped various envelopes, gates, doors and boxes.

 

[roger]

n.

And no amount of bluster about magic or wormholes changes that.

Rolfe.

bluster? That's pretty hostile. I purposefully chose silly things for fun.

It's just a math puzzle with some poorly chosen wording. No need to get personal.

 

[Rolfe]

Well, apologies if I mistook fun for sniping. The point is, there are a number of perfectly reasonable assumptions which can be made. Wormholes and magic aren't in that category, but showing that some very silly assumptions are possible doesn't prove that only one set of assumptions is sensible.

Rolfe.

 

[Rolfe]

It doesn't say that anywhere, does it?

And I think it is far from obvious: The show works and brings in advertising money *because* a lot of prices are given away as well as a lot of prices being lost by people who had them or could have had them.

So Monty's intend would be to give away some cars, but not too many and not too often.

There used to be a similar show in Germany. Many years later the host was on a different show, demonstrating how he could easily steer people to do what he wanted....

That's where imagining a computer or mechanical device has advantages when posing the question merely as a mathematical puzzle. Once you start trying to second-guess how Monty might be trying to manipulate the contestants (as opposed to following a predetermined rule), you never end.

And I should swear off opening threads that have the words "Monty Hall" in the title.

Rolfe.

 

[roger]

Well put. If the problem is formulated so as to convey that Monty must open a second door, and that this act must reveal a goat, it's pretty simple to show that switching doubles your chance of winning.

What tends to happen though is that the problem is not formulated so rigorously, and when this obvious trusim is explained, someone then says, "oh but you're assuming he has to show you a goat, that wasn't specified." And once you start down that road, the possibility that he might not have to open a second door at all (and may choose to do so with unknown motives), or could choose to reveal the car (and so prevent you having the opportunity to switch), then gets dragged in

Two things about that. While it is simple that in the first case switching increases your odds, people object strenuously against that interpretation given all the rules we understand to apply. It's that part that makes it a good puzzle.

Second, despite your protestations, every puzzle has unstated assumptions. "A train leaves a station and travels 30mph". Instant acceleration? No variation in speed? Track length is measured including local topography (rolling hills and the like)? Of course, these puzzles are very common to us, and we know how to select the right assumptions.

In all the arguments about variations on interpretation of the Montal Hall problem, I haven't seen an alternative selection of rules that makes the problem 'interesting'. Monty selects randomly? How would that be a 'puzzle' - every last person would get the answer correct (switching conveys no advantage). Monty is evil? Monty also doesn't know what is behind the doors? You are assuming pretty significant things not stated in the problem, and you end up with a puzzle whose answer is either indeterminate or trivial. To me, part of a puzzle is figuring that sort of thing out. Including acceleration in the train problem, when we do not know the acceleration profile of the train renders it insolvable. Thus, we conclude, acceleration was not intended to be included in the problem (note this is not 'bluster' - every ninth grader exposed to physics is going to ask the question about acceleration).

We all (I think) recognize the common expression of the puzzle was ambiguous. I still think that if you think about puzzle construction, and are good, you can puzzle out what the correct assumptions are. As you point out, if you don't assume specific things, the puzzle quickly becomes unsolvable. As it turns out, even with the wording of the puzzle as it stands, everyone has figured out what the meaning of the puzzle must have been, because that is the only way it makes sense as a puzzle.

 

[roger]

Well, apologies if I mistook fun for sniping. The point is, there are a number of perfectly reasonable assumptions which can be made. Wormholes and magic aren't in that category, but showing that some very silly assumptions are possible doesn't prove that only one set of assumptions is sensible.

Rolfe.

I didn't see this response when I posted post #135. As I explain there, I don't think there are perfectly reasonable alternative assumptions that can be made. Everyone and their brother has figured out that the puzzle requires assumptions about: uniform distribution, Monty knows what is behind the door, Monty is not evil, Monty always opens a door with a goat, etc. Change the assumptions and the puzzle is either unsolvable or the answer is trivial.

 

[Rolfe]

We all (I think) recognize the common expression of the puzzle was ambiguous. I still think that if you think about puzzle construction, and are good, you can puzzle out what the correct assumptions are. As you point out, if you don't assume specific things, the puzzle quickly becomes unsolvable. As it turns out, even with the wording of the puzzle as it stands, everyone has figured out what the meaning of the puzzle must have been, because that is the only way it makes sense as a puzzle.

I'm inclined to agree with you, actually. It's just that many many people do not appear to see it that way. It's pretty much de rigeur for the question of whether or not Monty is obliged to reveal a goat when he opens the second door to be raised when discussing the problem. Usually, I have to say, in an attempt to show the smart-aleck who explained that (in the usual way the problem is understood) switching doubles your chances of winning.

I'm not really familiar with what all this stuff is about Vos Savant, but I think this is what it relates to - that she explained the answer adhering to the obvious assumptions, and then got roasted as being wrong, or stupid, or something, by people who decided that these assumptions were not inherent in the problem as presented.

Rolfe.

 

[roger]

I'm inclined to agree with you, actually. It's just that many many people do not appear to see it that way. It's pretty much de rigeur for the question of whether or not Monty is obliged to reveal a goat when he opens the second door to be raised when discussing the problem. Usually, I have to say, in an attempt to show the smart-aleck who explained that (in the usual way the problem is understood) switching doubles your chances of winning.

I'm not really familiar with what all this stuff is about Vos Savant, but I think this is what it relates to - that she explained the answer adhering to the obvious assumptions, and then got roasted as being wrong, or stupid, or something, by people who decided that these assumptions were not inherent in the problem as presented.

Rolfe.

Well, the wikipedia article is actually well written, with dozens of references to peer reviewed papers and the like. But, if you want to read the specifics on the vos Savant issue, here it is.

Many PhDs in mathematics, from obscure places like MIT )), understanding the assumptions correctly, disagreed with the argument that you should switch. It's a very non-intuitive problem, and got very heated debate. Of course, a subset of the debate was which assumptions did we make, and which are best to make, but my point in this thread is that is not the interesting part of the problem. It's interesting (to me, I grant) that even when all the assumptions are nailed down (wormholes nonwithstanding) very smart and educated people still disagree. Read the comments to her article - PhDs, grad students in mathmatics, all thought she was wrong, and continued to think so until they actually carried out an experiment based on what we all agree are the 'correct' assumptions for the 2/3 answer.

I'll selectively quote:

And a very small percentage of readers feel convinced that the furor is resulting from people not realizing that the host is opening a losing door on purpose. (But they haven't read my mail! The great majority of people understand the conditions perfectly.)

In other words, people made the correct assumptions, even ran the experiment, and still disagreed with what to us is the obvious solution.

 

[humber]

But again, where is it said that we should presume this?

I don't say that you should, but that is the intent of the writer of the question. For what other reason would the problem be posed as a game show? It is generally accepted that Monty works to avoid giving the prize away, and may even mislead the contestant. His intent is subsumed by the intent of the show's producers to provide entertainment. I think that perhaps applies to Parade's reason for publishing the puzzle, though not necessarily Vos Savant's intent

If he does want to avoid giving the car away, then obviously he only offers the switch if the contestant has already picked the car.

He can do that or not, if the contestant picks the car, because there is always one goat. So it could be said his intent is to show the contestant a goat.

Which then becomes counterproductive once enough iterations are in the public domain for this strategy to be deduced.

Yes, but I don't think actual games are reduced to one prize and two identical alternatives. One may be cash, or a holiday for example. Perhaps the car has a higher monetary value, but a holiday may be preferred if that allows a week away from the spouse.
If it were to become known that the prize were more likely to be behind the first curtain, I suppose the show would react to that too.

It seems fairly obvious that the problem as originally posed, is intended to cover the situation where Monty always opens a second door and always reveals a car, leading to the brain-teasing needed to understand that in that situation switching doubles your chances of winning. The problem lies in the puzzle not being rigorously enough defined.

As hgc said, he always reveals a goat.
But, I agree, remove the trimmings and it's just a simple example of conditional probability.

I've seen better attempts where it's posed not as a human choosing which doors to open, but an unthinking mechanical system which will always open a second door and always reveal a goat. That way you can get to the "yes, switching will double your chances of winning" answer with a lot less angst.

But that machine must follow an algorithm that mimics Monty's action of always revealing a goat.

I'm not t all familiar with Marilyn Vos savant in this context, though I've seen the name mentioned in this context. Did she go for unwarranted assumptions, or what? (As I understand the usual IQ scales, they max upt somewhere in the 170s, so I don't think I believe in anyone with an alleged IQ of 228.)
Rolfe.

There is a website dedicated to her errors, http://wiki.wiskit.com/marilyn,
"Marilyn is Wrong", but is not currently accessible.

I.Q. is a vague idea, and the results much disputed, but she does claim 228.

"Recently we had an e-mail inquiring where in the Guinness Book of Records Marilyn is listed. The following was reported to Herb Weiner by "Citeon:"
She is listed on page 26 of the 1989 Edition. It says "...as a ten year old achieved a ceiling score for 23-year-olds, thus giving her an IQ of 228." Her record was retired to the guinness hall of fame because scores that high are no longer possible due to a lowering of test score ceilings."

 

[marplots]

In all the arguments about variations on interpretation of the Montal Hall problem, I haven't seen an alternative selection of rules that makes the problem 'interesting'. Monty selects randomly? How would that be a 'puzzle' - every last person would get the answer correct (switching conveys no advantage). Monty is evil? Monty also doesn't know what is behind the doors? You are assuming pretty significant things not stated in the problem, and you end up with a puzzle whose answer is either indeterminate or trivial.

So we agree you should switch if Monty knows and reveals a goat and the advantage is 2/3 to you. We also agree that a random choice by Mr. Hall reduces that to 1/2. So here's a variation that might have legs.

Monty doesn't want to give away where the car is so he purposefully doesn't look ahead of time. You pick and he secretly flips a coin. He then peeks behind the door is he is about to reveal.

If he sees a car, he says, "You picked a goat, so I'm going to reset and give you another chance." The three items are remixed (so neither you or Monty know what's what) and the procedure is repeated. This continues until Monty's peek shows him a goat and he opens that door.

Under these circumstances, should you switch or not switch or does it matter? If your odds are reduced with this procedure instead of the regular one, isn't it counter-intuitive that him showing you your choice was wrong sometimes actually hurts your chances?