Monty Hall Problem

[CurtC]

Maybe, if we tried to apply game theory, we could get a more general solution (although under game theory there may not be a stable solution).

Yes, there is a stable solution. Assuming, in game theory style, that Monty wants to hang on to his prize and you want to win it, then Monty would never offer the chance to switch if you picked the wrong door initially - he would just take his prize and your turn is over. Now lets's say that he sometimes offers the switch when you picked the right door - you, the player, would know that he is only offering you the choice because you picked the right door initially, so you would never switch. Since you never would switch, Monty, in this game theory world, would never bother to even offer it.

In theory, practice is the same as theory. In practice, it isn't. The difference in this case is that Monty wants an exciting game show, and wants to give prizes away sometimes. He would sometimes offer the switch, and sometimes wouldn't. The problem statement in the OP does not constrain him to always offering it, so we have no way of knowing what criteria he uses to decide whether to offer the switch, so the problem statement has no solution.

It looks like we have three holdouts here, maintaining that his motivations don't matter, that everything you need to know is that it was offered this time. This is wrong. You do need to know his motivation for why he offered you the switch. I first heard this problem 15 years ago, and have been active in debating it for most of those. It took me a while too to realize that his motivations matter, and that the problem can't be solved as it's stated in the OP. Guys, re-read the explanations. This is a settled matter. And TeaBag420, you really don't come across too well when you're insulting, belligerent, and wrong.

 

[Paul C. Anagnostopoulos]

1) If Monty only reveals a door after the contestent choses the car, then it means you have selected the car and chosing means you will lose

2) If Monty only reveals a door after the contestant choses wrong, then it means you have selected the wrong door and should switch.

3) If Monty always reveals doors when someone picks one, or does it indiscriminately, then switching means you will win 2/3 of the time

Sorry, I didn't pay enough attention to these possibilities. However, I see nothing in the wording of the problem that would make me consider the first two scenarios. There is no hint of a conditional nature to the problem. It simply says I pick and door and Monty reveals a goat, no ifs, ands, or buts.

Here's a bunch of skeptics reading a child's word problem:

S1: Okay, it says "Johnny has 7 apples and gives 3 to his best friend Sally. How many apples does he have left?"

S2: Hmm. There's an obvious answer, but how do we know that one of his remaining apples isn't made of wax? Then he would only have 3 real apples left.

S1: Or he could give 3 real apples and one wax apple to Sally. That would still be "giving 3 apples," but now he would only have 3 apples left.

S2: How do we know that he gives her 3 apples only if she doesn't already have apples? If she does, he would have all 7 left.

S1: Yes, and perhaps Sally refuses apples if Johnny is wearing a red shirt. Then he'd still have all 7 apples.

~~ Paul

P.S.: I'm all for clarifying the wording.

 

[Yaotl]

Sorry, I didn't pay enough attention to these possibilities. However, I see nothing in the wording of the problem that would make me consider the first two scenarios. There is no hint of a conditional nature to the problem. It simply says I pick and door and Monty reveals a goat, no ifs, ands, or buts.

Here's a bunch of skeptics reading a child's word problem:

S1: Okay, it says "Johnny has 7 apples and gives 3 to his best friend Sally. How many apples does he have left?"

S2: Hmm. There's an obvious answer, but how do we know that one of his remaining apples isn't made of wax? Then he would only have 3 real apples left.

S1: Or he could give 3 real apples and one wax apple to Sally. That would still be "giving 3 apples," but now he would only have 3 apples left.

S2: How do we know that he gives her 3 apples only if she doesn't already have apples? If she does, he would have all 7 left.

S1: Yes, and perhaps Sally refuses apples if Johnny is wearing a red shirt. Then he'd still have all 7 apples.

~~ Paul

P.S.: I'm all for clarifying the wording.

That also doesn't clarify "give". Give permanently or give to just hold. Then it begs the question of what "have" means in this sentence. Have altogether or just have in his immediate possession. How do we know if he's not stashing more apples somewhere else? Do those count?

This is fun

 

[pgwenthold]

Sorry, I didn't pay enough attention to these possibilities. However, I see nothing in the wording of the problem that would make me consider the first two scenarios.

But how do you rule them out?

There is nothing in the original wording that would make you consider them, but there is nothing in the wording that prevents them, either.

 

[Cabbage]

Well, since no one seems to be biting on the example I gave, where the spectator makes a side bet with you (if the person switches and wins, you win the side bet; if the person switches and loses, you lose the side bet), I'll go ahead and finish it off, with the point I was going to make.

Let's say you decide to take the bet. I'm under the impression that everyone here, even those that claim Monty's motivations are irrelevant, would agree that this is a chancy bet. Whether you win or lose will depend greatly on how the operator plays the shell game in the long run, and whether or not he offers the switch consistently (and for clarity, I will state again--this shell game is a fair game, and the player can certainly win by finding the pea. Neither the operator can cheat (by not placing the pea under any shell), nor can the player cheat (by catching a glimpse of where the pea is, or following it in the shuffle)). This shell game is in every sense probabilistically equivalent to the Monty Hall game, with the stipulation that Monty may or may not always offer the switch.

Now that you've taken the bet, the next person comes up to play the shell game. You and the gentleman with whom you have made the wager wait to see what happens.

The contestant makes his selection, the shell game man reveals an empty shell, and offers the contestant the opportunity to switch.

Now, according to those who claim Monty's motivation makes no difference, the contestant believes his chances of winning will be 2/3 if he switches, and 1/3 if he doesn't switch. After all, this is only one game, and it's identical in every sense to the Monty Hall cars and goats game.

On the other hand, you, having made the side bet with the spectator, realize is not so clear as 2/3 vs. 1/3. You still have yet to learn how the shell game operator is operating--does he offer the switch consistently or not? Having seen the game only twice, it's far too early to tell.

So here we have two different people (you, having made the side bet, and the contestant, currently playing the game) judging the probabilities differently! Furthermore, you are both making those judgments with the exact same knowledge. The only difference between the two of you is that you have somewhat different bets riding on the outcome.

Is it possible for both people, with the exact same knowledge of the events, to come to different conclusions about the probabilities, and have both of them be correct? I don't see how.

 

[Paul C. Anagnostopoulos]

I would rule them out on the basis that these are fun math problems for normal people designed to point out that our instincts about probability are often incorrect. Also on the basis that it mentions absolutely nothing about any conditions on Monty's opening the goat door. If there were any conditions, they would have to be stated so we would have the complete problem. No conditions stated, no conditions implied.

Is that mathematically rigorous enough for ya?

~~ Paul

 

[rppa]

Sorry, I didn't pay enough attention to these possibilities. However, I see nothing in the wording of the problem that would make me consider the first two scenarios. There is no hint of a conditional nature to the problem.

Sorry, but I disagree. And I'm a hardcore two-thirder.

It simply says I pick and door and Monty reveals a goat, no ifs, ands, or buts.

That's right. But in order to assess expectation values, you need to come up with a probability model for how Monty will behave under all possible scenarios, and all you have is a single data point: whatever prize I have behind my door, Monty is showing me a goat.

You've got to make SOME assumption about his behavior, or there's no way to calculate the probabilities. There is absolutely more than a hint of a conditional nature to the problem. The assumptions about the conditional nature are key to your calculation of expectation value.

However, somebody (Robin?) has done a very nice job of summarizing several different hypotheses about what the single observation we have might mean in general. And it bears out what I read in at least one website: The only time it doesn't pay to switch is if I think "Monty shows me a goat" implies "I have the car already".

Here's a bunch of skeptics reading a child's word problem:

Sorry, this is a strawman. As soon as you start working out the potential payoffs, you are going to make assumptions about Monty's rules of operation. It's impossible to calculate otherwise. The information stated is not sufficient to answer the question. Assumptions are needed.

What makes you think the information given is sufficient to know what Monty does under all circumstances?

 

[Robin]

rppa
You've got to make SOME assumption about his behavior, or there's no way to calculate the probabilities.

This is absolutely right. You could make a meal of what you are not told. Are you a contestant on the game show? Or is this an "ask the audience" segment where your choice gives you no advantage or disadvantage? What is the nature of the show, what are the rules? The car might be an old clunker and the donkey's might have $100,000 under their saddles.

Or it might be theatre sports where the advantage depends in how well you play the role.

 

[hgc]

This is absolutely right. You could make a meal of what you are not told. Are you a contestant on the game show? Or is this an "ask the audience" segment where your choice gives you no advantage or disadvantage? What is the nature of the show, what are the rules? The car might be an old clunker and the donkey's might have $100,000 under their saddles.

Or it might be theatre sports where the advantage depends in how well you play the role.

Now it's everything AND the kitchen sink, eh? Please tell me how the relative value of the car and the donkeys affects the outcome.

 

[Robin]

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Originally posted by Paul C. Anagnostopoulos
Sorry, I didn't pay enough attention to these possibilities. However, I see nothing in the wording of the problem that would make me consider the first two scenarios.
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Originally posted by pgwenthold:
But how do you rule them out?

There is nothing in the original wording that would make you consider them, but there is nothing in the wording that prevents them, either.
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It depends entirely on how you read it. It is like one of those pictures that looks like a young woman to some and an old woman to others.

If you, as I did, read it as "this is how the game is played" then the reasonable interpretation is that the host always offers the switch.

If you read it, as others have, as "this is a specific scenario within the game" then you don't have enough information to know.

The wording supports both readings.

However I still hold out that the motivations of the host (or director or whatever) is not the key question.

The key question is "does the host always offer a switch?".

Without that piece of information the problem does not have an answer.

 

[Yaotl]

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?

It's

What is the probability that you will get the car by switching, or by staying?

Nowhere does it state what is the probably overall in the game, just this instance. Whether he does it later on or not for another contestant or the audience or whatever is irrelevant. You have to figure out your odds right then and there.

 

[hgc]

Re: Re: Monty Hall Problem

It's

Nowhere does it state what is the probably overall in the game, just this instance. Whether he does it later on or not for another contestant or the audience or whatever is irrelevant. You have to figure out your odds right then and there.

Agreed. Did you think I thought anything different?

 

[Yaotl]

Re: Re: Re: Monty Hall Problem

Agreed. Did you think I thought anything different?

Nope, I just wanted to restate what you said since I doubt anyone's looking at the original problem anymore. What with all the assumptions being bandied about and all.

 

[hgc]

Re: Re: Re: Re: Monty Hall Problem

Nope, I just wanted to restate what you said since I doubt anyone's looking at the original problem anymore. What with all the assumptions being bandied about and all.

Thanks. I can see that the question needs to be reworded to exclude ambiguity about Monty's reason for revealing the donkey and offering the switch, but there is a lot of other irrelevancy flying around.

 

[CurtC]

hgc wrote:
Thanks. I can see that the question needs to be reworded to exclude ambiguity about Monty's reason for revealing the donkey and offering the switch...

Thank you. That was my point. It needs to be specified, in the problem description, that Monty is constrained to offer you the choice to switch no matter what you picked or what mood he was in at the time.

 

[hgc]

Thank you. That was my point. It needs to be specified, in the problem description, that Monty is constrained to offer you the choice to switch no matter what you picked or what mood he was in at the time.

And not that Monty's intentions don't make for interesting discussion, but it won't work too well in a bar bet situation.

 

[Robin]

Now it's everything AND the kitchen sink, eh? Please tell me how the relative value of the car and the donkeys affects the outcome.

The wording of the original MHP (which was brought into play by epepke and gnome) said "Is it to your advantage to switch?". Clearly in this case the relative retail values are very relevant.

OK so the OP said "what is the probability that you will get the car by switching?". You still have a problem - it is not stated that you will get the car by guessing which door it is behind.

Maybe the point of the game is to eliminate prizes you don't want. Maybe like "Sale of the Century" picking the car but in the wrong order means you don't get it.

The point I am making is that you have to assume something, so what we want to know is what is it reasonable to assume?.

 

[Robin]

So what is actually being debated? Is it this?

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?

If so then the answer is easy. The answer is that there is not enough information to calculate a probability. There is not even enough information to state whether or not Monty's intentions are relevant. Even when making reasonable assumptions.

It is silly to spend so much time debating when we have not even decided what is being debated.

 

[hgc]

The wording of the original MHP (which was brought into play by epepke and gnome) said "Is it to your advantage to switch?". Clearly in this case the relative retail values are very relevant.

OK so the OP said "what is the probability that you will get the car by switching?". You still have a problem - it is not stated that you will get the car by guessing which door it is behind.

Maybe the point of the game is to eliminate prizes you don't want. Maybe like "Sale of the Century" picking the car but in the wrong order means you don't get it.

The point I am making is that you have to assume something, so what we want to know is what is it reasonable to assume?.

Yes, when people communicate via human language, they make assumptions about meanings of words and phrases based on shared cultural experiences. If you want to engage in pedantic nitpickery you can uncover all kinds of bogus ambiguity.

Who didn't know, unquestionably, that the picking of the door was for receiving the prize behind that door? Raise you hands. I can understand that some puzzles are constructed with tricky loopholes hidden in the precise construct of the question. No reason to think that this is such a puzzle. This is a puzzle about probability.

 

[Cabbage]

Re: Re: Monty Hall Problem

Nowhere does it state what is the probably overall in the game, just this instance. Whether he does it later on or not for another contestant or the audience or whatever is irrelevant. You have to figure out your odds right then and there.

I'm reading this as implying you think these long term odds (as more and more contestants play the game) are somehow different from the short term odds (when you play a single game).

How can that be so?

Read my example where two bets are going on--One being the game as it is being currently played by a contestant (short term, only one game), while the other is a a wager made by a person on the outcome of the games (over some long term) whenever the current contestant switches.

How could you analyze such a situation in a way that gives different probabilities from the two different perspectives?