Monty Hall Problem

[Rolfe]

I hope you meant to say "the answer is 1/3"

I assumed this was "department of we know what he meant!
























In marked contrast to the originator of the "Monty Hall" puzzle.

Rolfe.

 

[Skeptical Greg]

I just found something interesting in Wikipedia. Marylin vos Savant, who may be considered to be one of the world's experts on the Monty Hall problem, wrote about the version where Monty chooses the door at random in her column in November 2006:

This difference can be demonstrated by contrasting the original problem with a variation that appeared in vos Savant's column in November 2006. In this version, Monty Hall forgets which door hides the car. He opens one of the doors at random and is relieved when a goat is revealed. Asked whether the contestant should switch, vos Savant correctly replied, "If the host is clueless, it makes no difference whether you stay or switch. If he knows, switch" (vos Savant, 2006).

That explanation seems non-sensical ..
If Monty forgets which door hides the car, how can him ' knowing ' be a factor ?

 

[Michael C]

That explanation seems non-sensical ..
If Monty forgets which door hides the car, how can him ' knowing ' be a factor ?

But that's the whole point: of course if Monty forgets which door hides the car, he no longer knows where the car is! In the first version Monty knows where the car is, in the second (random choice) he doesn't know where the car is. The important point is that whether he knows or not makes a difference to the outcome.

 

[Skeptical Greg]

Your reference clearly says, vos Savant is asked if the contestant should switch when the host has forgotten... She says ' switch if he knows ' ...

The example is self contradictory ..

 

[AntiTelharsic]

YOU assume the problem to be identical if repeated. However, there is no basis for that assumption. As written in the OP, it only describes a single instance.

If "The Monty Hall Problem" is "Monty presents 3 doors; behind one is a car, and behind the other two are goats. You pick a door. But before Monty opens that door, he reveals behind another door that there is a goat. He then gives you the opportunity to switch your choice. What is the probability that you will get the car by switching, or by staying?" then that's what happens the first time I take part in "The Monty Hall Problem". It's also what happens the next time I take part in "The Monty Hall Problem" unless someone re-defined what "The Monty Hall Problem" is in the meantime. You're free to assume that it might change, and that Monty might reveal a car, but if you do you must change the statement of "The Monty Hall Problem", removing the bit where he reveals a goat, and you are then no longer talking about "The Monty Hall Problem" as it was originally stated.

The statement of the problem was not "he reveals what's behind a random door, which turns out to be a goat". The statement of the problem was "he reveals behind another door that there is a goat." That's how the problem works -- he reveals a goat. Change that, and you change the problem.

You say there's no information either way, but there's an explicit statement that he reveals a goat, and it's still there every time someone looks at the problem. You're adding information if you're assuming there are hidden processes, such as random determination of the revealed door, in operation here.

This thread looks like the Monty Python problem. "I'd like to have an argument, please." "Do you want to have the full argument, or just 13 pages?"

But seriously.. discussing variants of the problem is great; let's just call them variants.

 

[CurtC]

If "The Monty Hall Problem" is "Monty presents 3 doors; behind one is a car, and behind the other two are goats. You pick a door. But before Monty opens that door, he reveals behind another door that there is a goat. He then gives you the opportunity to switch your choice. What is the probability that you will get the car by switching, or by staying?" then that's what happens the first time I take part in "The Monty Hall Problem". It's also what happens the next time I take part in "The Monty Hall Problem" unless someone re-defined what "The Monty Hall Problem" is in the meantime.

I think the misunderstanding here has to be around the interpretation of the words used in the OP's statement of the puzzle. When he said "But before Monty opens that door, he reveals behind another door that there is a goat. He then gives you the opportunity to switch your choice.," you are taking that to mean that you are playing a game where Monty will always give you three doors to choose from, then after you make a choice, will always show you a non-prize door and give you an option to switch.

Again (and again), what we're saying is that the OP can be interpreted to not be specific enough - a game where Monty gives you a choice of three doors, then if you picked the prize on your first guess will then show a non-prize door and give you the chance to switch, this game is consistent with the wording in the OP if you read his words only as a description of a one-time situation that you found yourself in.

Here's the description of this issue found at the Wikipedia page:

Some of the controversy was because the Parade version of the problem is technically ambiguous since it leaves certain aspects of the host's behavior unstated, for example whether the host must open a door and must make the offer to switch. Variants of the problem involving these and other assumptions have been published in the mathematical literature.

The issue is that the wording of the OP in this thread also leaves aspects of the host's behavior unstated, or at best can be interpreted ambiguously about the host's behavior.

 

[AntiTelharsic]

Oh, yeah, it certainly can be interpreted ambiguously, but the simplest interpretation is the standard one. Again, if you want to assume that the host has decision-making processes which aren't specified, you're adding information to the specification of the problem. If the host's methodology is unstated, it's simplest to assume he doesn't have one and does precisely what the problem said: reveals a goat.

 

[jsfisher]

If "The Monty Hall Problem" is "Monty presents 3 doors; behind one is a car, and behind the other two are goats. You pick a door. But before Monty opens that door, he reveals behind another door that there is a goat. He then gives you the opportunity to switch your choice. What is the probability that you will get the car by switching, or by staying?" then that's what happens the first time I take part in "The Monty Hall Problem". It's also what happens the next time I take part in "The Monty Hall Problem" unless someone re-defined what "The Monty Hall Problem" is in the meantime.

Yes, and yes again.

And if on trial number three, Monty doesn't bother opening any door, then that is not "The Monty Hall Problem". True enough. Then, on the forth trial, Monty again opens a door to reveal a goat and gives you an option to change your choice of door. Then this is "The Monty Hall Problem" again.

"The Monty Hall Problem" is whether you should change your choice when faced with the sequence of events you stated. It is a conditional probability problem: What's the probability the remaining door is a better choice given that Monty has shown us a goat.

Unfortunately, all the Monty Hall Problem statement provided was a sequence of events, not any binding guarantee that that sequence had to happen. We do not have enough information to compute the conditional probability.

You're free to assume that it might change, and that Monty might reveal a car, but if you do you must change the statement of "The Monty Hall Problem", removing the bit where he reveals a goat, and you are then no longer talking about "The Monty Hall Problem" as it was originally stated.

No, not at all. We were only told in the opening post what did happen, not what could have happened. Had a different sequence of events occurred, then, no, it would not have been "The Monty Hall Problem", but that doesn't change that it still may have been a possible outcome.

The statement of the problem was not "he reveals what's behind a random door, which turns out to be a goat". The statement of the problem was "he reveals behind another door that there is a goat." That's how the problem works -- he reveals a goat. Change that, and you change the problem.

Ok, so he revealed a goat. You have no information to indicate he intended to reveal a goat. You have no information to indicate he would have opened any door at all had we selected a different door. We only know he did open a door after we chose a door, and behind the door was a goat.

You say there's no information either way, but there's an explicit statement that he reveals a goat, and it's still there every time someone looks at the problem. You're adding information if you're assuming there are hidden processes, such as random determination of the revealed door, in operation here.

The explicit statement is that a goat was revealed. It was not that you pick a door then Monty will reveal a goat. You only know what has happened, not what had to happen.

 

[AntiTelharsic]

If I were posing the problem to someone, and Monty's behavior were variable, I'd be sure to point that out in my description of the problem. The fact that it was not pointed out implies (or, at least, strongly suggests) that such considerations are irrelevant to the problem.

I'm not going to say "If I flip a coin, what is the probability of getting heads?" and then go "Hah! You lose! It was a loaded coin!" when you answer "1/2". The simplest, most reasonable assumption is that the coin is fair. If the coin were unbalanced, or had two heads, or if I would be using different kinds of unfair coins on repetitions of the problem -- if there were any features of my problem which would affect your computation of the answer -- there wouldn't be any sense in my not telling you, because you'd be working on a different problem from the one which I intended to pose.

Yes, you could complain that my wording was ambiguous and allows for all sorts of dirty tricks I could do with my coins, but if you assume that I'd actually like for you to be able to solve the problem I posed to you, you wouldn't bother.

 

[CurtC]

Again, if you want to assume that the host has decision-making processes which aren't specified, you're adding information to the specification of the problem.

Au contraire, I'm not adding anything, it's you who is doing so. You're adding the assumption that Monty is playing by some unspecified rules where he is forced to always show you a non-prize door. I'm simply pointing out that your assumed rules are not part of the puzzle statement.

I used to watch the TV show when I was younger, and one thing for certain, Monty was not bound by your assumption in real life. I point this out because you were appealing to the reasonableness of the assumption, when it's not reasonable at all.

 

[AntiTelharsic]

I used to watch the TV show when I was younger, and one thing for certain, Monty was not bound by your assumption in real life.

The question is not about the TV show. It is a self-contained problem involving characters and situations from a TV show.

When you're given the statement of the problem, and that's all the information you're given, then it is reasonable to assume that you've been given enough information to solve the problem. If you assume that part of the problem is that Monty's behavior may deviate from that stated, then you don't have enough information to solve the problem. The problem would not be posed without giving you enough information to solve it.

 

[AntiTelharsic]

Au contraire, I'm not adding anything, it's you who is doing so.

You're adding the requirement that we understand some unspecified decision-making process for Monty.

You're adding the assumption that Monty is playing by some unspecified rules where he is forced to always show you a non-prize door. I'm simply pointing out that your assumed rules are not part of the puzzle statement.

The puzzle's statement says he reveals a goat. It said it the first time I read it, the second time I read it, and indeed on all subsequent readings. Every time I go to solve the puzzle, there it is: he reveals a goat. Lacking any statement that he may one day not reveal a goat, I can only assume that he will reveal a goat the next time I try to solve the puzzle.

If you're supposed to be able to solve the puzzle with only the information given, then you assume that he will always reveal the goat. If you don't assume that he will always reveal the goat, then you need more information and you can't solve the puzzle. I think it was the OP's intent that the puzzle as stated be solvable. You're free, of course, to think it was a trick.

 

[jsfisher]

The question is not about the TV show. It is a self-contained problem involving characters and situations from a TV show.

When you're given the statement of the problem, and that's all the information you're given, then it is reasonable to assume that you've been given enough information to solve the problem. If you assume that part of the problem is that Monty's behavior may deviate from that stated, then you don't have enough information to solve the problem. The problem would not be posed without giving you enough information to solve it.

For the coin flip question you posed earlier, you said to assume it to be a fair coin (and a fair flip, too) would be reasonable. Your general point was (I think) that given an imprecise problem statement, it is entirely appropriate to make minimal, reasonable assumptions to make the problem unambiguous and solvable.

Ok, fair enough. I see the point. However, in the Monty Hall Problem, I think you have over-reached minimal and reasonable to make your assumptions.

Why is it more appropriate to assume Monty will open a door and behind the door will be a goat than to assume Monty will open a door which in this case has a goat behind it?

Consider this slightly changed coin flip question: What is the probability a flipped coin will come up heads immediately after it being flipped and came up heads?

I'd say your original fair coin / fair flip assumptions were still appropriate, so the answer should be 1/2. The coin coming up heads for the first flip is just the goat that happened to be behind the door. So, wouldn't Monty picks a door at random be at least as valid an assumption as the one you proposed?

 

[AntiTelharsic]

Your general point was (I think) that given an imprecise problem statement, it is entirely appropriate to make minimal, reasonable assumptions to make the problem unambiguous and solvable.

You put it better than I've been able to

Why is it more appropriate to assume Monty will open a door and behind the door will be a goat than to assume Monty will open a door which in this case has a goat behind it?

Because you're supposed to be able to solve the problem. If you don't make that assumption, then you need more information before you can solve it; specifically, you need to know how Monty decides which door to open.

I'd say your original fair coin / fair flip assumptions were still appropriate, so the answer should be 1/2. The coin coming up heads for the first flip is just the goat that happened to be behind the door. So, wouldn't Monty picks a door at random be at least as valid an assumption as the one you proposed?

With coin tosses it's generally understood that there is a random process with a well-known distribution. I think any sort of variable process for Monty, however, random or otherwise, is a more complex situation than him simply acting as described. If it's variable, how does it vary? If it varies randomly, what's the distribution? Too many questions to be asking when there's a way to solve the problem by making the least complex assumption: that he will just act as described.

If you were writing it as an exam question for students, and you intended for Monty's behavior to vary, then you would include information about how it would vary. It's the only fair way to do it. If you received the OP's statement in an exam, how would you solve it?

 

[CapelDodger]

Look, people, this isn't so hard to grasp. The Monty Hall problem is not some horror movie about a game show host out to cheap unsuspecting players. It's a hypothetical scenario, specifically designed to show how unintuitive probability theory can be by presenting what seems to be a simple problem and showing that the 'gut' solution is incorrect.

My single contribution to this thread is :

What Mobyseven said.

I'm only contributing because it came up across the card-table this evening, and the guy that didn't initally get it did get it the moment it was explained. Serious card-players understand artifical systems with a finite number of discrete elements and a few simple rules about how they relate to each other.

 

[ddt]

No, that doesn't imply that this is how he always behaves. Consider these two problems:

1. I throw a coin. It comes up heads. Can I deduce that there is anything unusual about the coin?

2. I throw a coin. It always comes up heads. Can I deduce that there is anything unusual about the coin?

I don't think this is a fair comparison. A coin flip, in such a puzzle, is a chance event. Anyone reading the puzzle assumes that. However, the description of Monty's behaviour is a deterministic description.

In any case, I am happy that the OP left the possibility of interpreting the problem in different ways, since it led to the discussion on whether the version where Monty always knows the location of the prize is different from the one where he chooses a door at random. The two versions are indeed different and have different solutions.

I agree that the OP - and Marilyn vos Savant's description in Parade in 1990 which generated so much publicity for it, which was nearly identical - leave indeed both interpretations open. However, I think my interpretation is the more usual one. I cite again the NYT article:

"The problem is not well-formed," Mr. Gardner said, "unless it makes clear that the host must always open an empty door and offer the switch. Otherwise, if the host is malevolent, he may open another door only when it's to his advantage to let the player switch, and the probability of being right by switching could be as low as zero." Mr. Gardner said the ambiguity could be eliminated if the host promised ahead of time to open another door and then offer a switch.

Ms. vos Savant acknowledged that the ambiguity did exist in her original statement. She said it was a minor assumption that should have been made obvious by her subsequent analyses, and that did not excuse her professorial critics. "I wouldn't have minded if they had raised that objection," she said Friday, "because it would mean they really understood the problem. But they never got beyond their first mistaken impression. That's what dismayed me."

In other words: all those angry letter writers who objected to her solution, did indeed follow my interpretation.

Why is it more appropriate to assume Monty will open a door and behind the door will be a goat than to assume Monty will open a door which in this case has a goat behind it?

Because it is thus stated. From the OP:

But before Monty opens that door, he reveals behind another door that there is a goat.

He'd be hard-pressed to reveal a goat behind the door which hides the car.

Or in from the wiki article:

and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat.

So whenever Monty opens a door, it is one with a goat behind it.

 

[Herzblut]

Set 1 - picks the car on the first guess, Monty then reveals a goat and offers the choice of switching. This group has 100 people.

Why? The group might only have 10 people. Or 250. Probability is not certainty.

 

[jsfisher]

Because it is thus stated. From the OP:

He'd be hard-pressed to reveal a goat behind the door which hides the car.

Here we disagree. It may be because of a subtlety of the language, or maybe just an ambiguity. My interpretation of the "he reveals behind another door that there is a goat" clause is that it presents two facts: Monty opened another door and there was a goat behind the other door.

Let's agree for the sake of discussion that Monty always opening a door is a minimal, reasonable assumption needed to make the problem solvable. If that is the only assumption made then switching doors is a 50% proposition. You'd have to make a second assumption to connect the two facts and require Monty open a goat door. (And then, it becomes a 67% proposition favoring switching.)

Are two assumptions (Monty always opens a door and it's always a goat door) more minimal and reasonable than just one assumption (Monty always opens a door)?

 

[Herzblut]

The puzzle's statement says he reveals a goat. It said it the first time I read it, the second time I read it, and indeed on all subsequent readings. Every time I go to solve the puzzle, there it is: he reveals a goat. Lacking any statement that he may one day not reveal a goat, I can only assume that he will reveal a goat the next time I try to solve the puzzle.

If you're supposed to be able to solve the puzzle with only the information given, then you assume that he will always reveal the goat. If you don't assume that he will always reveal the goat, then you need more information and you can't solve the puzzle. I think it was the OP's intent that the puzzle as stated be solvable.

That's my understanding of the problem. Monty reveals a goat after the player has made his choice. Whether the game is played once, twice or 1000 times, in each trial the player makes his choice and then a goat is revealed. Any other case (car, chimpanzee, Britney Spears..) is undefined and must be speculated about, because the description does not consider those options. I don't either.

 

[Herzblut]

Let's agree for the sake of discussion that Monty always opening a door is a minimal, reasonable assumption needed to make the problem solvable. If that is the only assumption made then switching doors is a 50% proposition.

The minimum problem solvable is the one I presented before, I think:

Monty opens a door and offers the player to stay or switch to another door of his choice.

Always switching then wins in 2/3 of the cases. Like the original game, it corresponds to the offer to either stick to the picked door or to switch to the two other doors. But it's of course a much less ingenious version.