Monty Hall Problem

[Cabbage]

You summed up your argument when you said "preludes to the OP". There are no preludes to the OP. You might as well say, "What if the Tooth Fairy offers Monty a really big bribe to throw the game?" The OP is the OP and you can string all the pearl necklaces you want on it, but it doesn't change.

This isn't really a fair comparison. Adding the comment about the tooth fairy skews the problem, and leaves it fundamentally changed.

However, the additions regarding the strategy of the host are actually relevant, since the original problem, as stated, leaves out vitally important information regarding the host's strategy. In other words, these comments clarify a previously vague problem.

 

[Cabbage]

And maybe I'm beating this point into the ground now, but here's another illustrative example.

Say you're playing a game of chess. Just after you make a particular move, you realize you've made a blunder--your opponent can now force checkmate in three moves. You cannot, however, take back this blundered move.

What is the probability that you will lose the game?

(This problem is not analagous to the original Monty Hall problem in a probabilistic sense, but it is analogous in the sense they are both vague).

Who can say what the probability is? If your opponent is an experienced grand master, you will most certainly lose the game. On the other hand, if your opponent is merely a beginner, it's quite possible he will miss the opportunity for mate, make his own blunder later on, costing himself the game. I have personally been in this situation, blundered into a losing position, yet still managed to take a victory simply because my opponent missed his opportunity.

The probability of you losing your chess game depends on the quality of your opponent.

Similarly, in the Monty Hall game, any advantage you may or may not gain by switching is unclear. It fully depends on Monty's strategy (or lack thereof, if you will), which is an unknown variable.

 

[TeaBag420]

Cabbage wrote:

"This isn't really a fair comparison. Adding the comment about the tooth fairy skews the problem, and leaves it fundamentally changed."

Godfrey Daniels, Cabbage. Adding anything skews the problem. That's the problem with adding to the problem, which has already been posted to this board.

 

[TeaBag420]

And maybe I'm beating this point into the ground now, but here's another illustrative example.

Say you're playing a game of chess. Just after you make a particular move, you realize you've made a blunder--your opponent can now force checkmate in three moves. You cannot, however, take back this blundered move.

What is the probability that you will lose the game?

(This problem is not analagous to the original Monty Hall problem in a probabilistic sense, but it is analogous in the sense they are both vague).

Who can say what the probability is? If your opponent is an experienced grand master, you will most certainly lose the game. On the other hand, if your opponent is merely a beginner, it's quite possible he will miss the opportunity for mate, make his own blunder later on, costing himself the game. I have personally been in this situation, blundered into a losing position, yet still managed to take a victory simply because my opponent missed his opportunity.

The probability of you losing your chess game depends on the quality of your opponent.

Similarly, in the Monty Hall game, any advantage you may or may not gain by switching is unclear. It fully depends on Monty's strategy (or lack thereof, if you will), which is an unknown variable.

I agree with everything you say above. Additionally, this is a quintessential example of problem changing. Problem changers are one step above kid touchers in my book. In the OP, you don't know and shouldn't care about Monty's strategy, but you know what he did.

(This problem is not analagous to the original Monty Hall problem in a probabilistic sense, but it is analogous in the sense they are both vague).

Game, set, match. Anything vague is analogous to the original Monty Hall problem in a way you think is important and/or relevant.

You go now. No trouble.

 

[Cabbage]

Cabbage wrote:

"This isn't really a fair comparison. Adding the comment about the tooth fairy skews the problem, and leaves it fundamentally changed."

Godfrey Daniels, Cabbage. Adding anything skews the problem. That's the problem with adding to the problem, which has already been posted to this board.

The addition doesn't skew the problem if its purpose is actually to clarify the problem. In fact, it clarifies a problem which was initially unsolvable.

 

[Cabbage]

I agree with everything you say above. Additionally, this is a quintessential example of problem changing. Problem changers are one step above kid touchers in my book.

Where did I change the problem? I merely pointed out the fact that having a single example of the game in which the host offers the switch is insufficient information to deduce what the host's strategy in the game may be. Your probability of winning or losing the game depends in a vital manner on that host's strategy.

 

[Paul C. Anagnostopoulos]

CurtC said:
Well, this is entertaining. So far, I think everyone gets it except Teabag420. Paul C. has some reservations, that even saying you don't know the host's strategy is making an assumption, which I think is being a little obtuse. The problem statement doesn't give us any information on the host's strategy, but his strategy would be necessary to know in order to solve for the probability. Saying "there's not enough information" is not making an assumption, it's admitting that we can't solve it unless we make an assumption beyond what the problem says.

My point is an informal one, because I agree that if this were on the National Math Test, clarity would be critical. I just don't see why we need to worry about conditions that are not explicitly stated in this problem, when considering it informally in order to make the point about peoples' probability intuitions. Isn't it reasonable to assume that, by default, unstated conditions do not exist? The point, after all, is to allow someone to solve the problem!

Also, folks, this problem statement is all over the Net, and sometimes it does make it clear that he always offers a choice:

http://www.remote.org/frederik/projects/ziege/

The candidate chooses one of the doors. But it is not opened; the host (who knows the location of the sports car) opens one of the other doors instead and shows a goat. The rules of the game, which are known to all participants, require the host to do this irrespective of the candidate's initial choice.

Of course, if we persist, this is still ambiguous. Monty may be required to offer a choice regardless of the player's initial selection, but it doesn't say he has to do it if the player is wearing red pants.

~~ Paul

 

[Drooper]

M I just don't see why we need to worry about conditions that are not explicitly stated in this problem,
~~ Paul

What? and spoil all the fun?

 

[BillHoyt]

My point is an informal one, because I agree that if this were on the National Math Test, clarity would be critical. I just don't see why we need to worry about conditions that are not explicitly stated in this problem, when considering it informally in order to make the point about peoples' probability intuitions. Isn't it reasonable to assume that, by default, unstated conditions do not exist? The point, after all, is to allow someone to solve the problem!

Also, folks, this problem statement is all over the Net, and sometimes it does make it clear that he always offers a choice:

http://www.remote.org/frederik/projects/ziege/



Of course, if we persist, this is still ambiguous. Monty may be required to offer a choice regardless of the player's initial selection, but it doesn't say he has to do it if the player is wearing red pants.

~~ Paul

No, no, no, Paul. Good grief! Can't you understand the absolute need to twist a simple probability problem into a game-theoretic one while introducing hidden assumptions about motivations, the exchange value of the US$ and how many speckled dwarves are hiding in Carol Merrill's mini-skirt?

 

[pgwenthold]

No, no, no, Paul. Good grief! Can't you understand the absolute need to twist a simple probability problem into a game-theoretic one while introducing hidden assumptions about motivations,

Any calculation of the probability requires some sort of assumption. The 2/3 probability that is the common answer requires the assumption that the option is allowed indiscriminately. If you assume that the option is only allowed when you have initially chosen the car, then the probabilty that you will win when you switch is 0. If you assume that the option is only allowed when you have chosen wrong, then the answer is 1. However, the short answer is that any answer requires an assumption. Since you don't have any information regarding his strategy, the question can't be answered absolutely.

It's actually very similar to geometry. The deniers sound like those defending Euclidian geometry, trying to claim that the 5th postulate is obvious. Others keep saying, yeah but it's not required. Within the constraints of the problem, you can create other scenerios where the probability is much different from 2/3. It all depends on your initial assumptions.

As Robin points out, without any assumptions, the probability cannot be calculated.

 

[Yaotl]

Good god (or gods), two more pages since I went home? I'm going to make a solemn vow never to discuss the MHP with anyone ever again.

 

[CurtC]

Paul and Bill, I think we all agree that the probability will be very different depending on whether the host always offers the switch, or offers it only if you picked right initially, etc., in other words, his strategy. And we all agree that if he was constrained to offer the switch without regard to your initial pick, that the answer is 2/3 in favor of switching.

Universal agreement on these, in my experience with this problem, is quite an accomplishment.

The only thing we seem to still disagree on is whether, in the problem statement as commonly stated, it's reasonable to assume that Monty's offer to switch would have been granted independent of your first guess. I have to admit that I can't understand the position that the assumption is warranted. Since you two are the ones making it, could you defend it? I'm curious how the reasoning would go. (I'm explicitly not inviting TeaBag420 to respond, because his m.o. so far has been to spew insults and exasperation with no substance.)

 

[Dr Adequate]

If Monty Hall had made a practice of offering the switch only when the first guess was right, that would have been a fairly poor strategy. And the answer would be "never switch". And no-one would be in the least interested in the problem.

The Monty Hall Problem is as stated. He always unveils one goat. He can't have a "strategy" as such. It doesn't matter whether this was true of the actual Monty Hall, or whether it's a scurrilous lie. Maths problems can be fictional.

And yes, the answer is that you should switch.

The shortest argument goes like this. When you picked, there was a 1/3 chance you were right first time. After the unveiling, there's still a 1/3 chance you were right first time --- nothing can change that. This means that there's a 2/3 chance that the car's behind the unselected drawn curtain.

 

[CurtC]

The Monty Hall Problem is as stated. He always unveils one goat.

Maybe you can point me to a statement of the problem where he always unveils one goat? I don't think I've ever seen one in the wild.

Let's go back to the street hustler version. Let's say it's a given that this is a fair shell game, that the hustler makes his money because he moves the shells so quickly that anyone has only a 1/3 chance of guessing correctly. You put down your money, make your pick, and then the hustler doesn't reveal your cup, but instead turns over an empty cup and offers you the choice to switch. Should you?

You'd be a fool to switch in this case, I hope that's obvious. And this problem is no different in substance from the Monty Hall problem. The only difference is what your assumed motivations for the game host are.

 

[Number Six]

I wonder what the original wording was in Ask Marilyn. I don't recall it saying that Monty always switches and also she fooled many people educated in the field of probability and if she had made clear that the switch would always be made then I think fewer people would have been fooled.

In general, I often hear the puzzle posed without it made clear that Monty always switches. It's just "You're on Let's Make
A Deal...you pick a door...Monty opens another door and shows you a goat and asks you if you want to change your choice...should you change your choice?" When it's worded that way there is no correct answer unless you make assumptions.

 

[BillHoyt]

This is just getting monumentally silly, folks. This is a probability problem, pure and simple, despite all the what-ifs, how-abouts, assumptions, lace, garters and speckled dwarves you try to toss in. Here is a clear statement, from the MathWorld site:

Assume that a room is equipped with three doors. Behind two are goats, and behind the third is a shiny new car. You are asked to pick a door, and will win whatever is behind it. Let's say you pick door 1. Before the door is opened, however, someone who knows what's behind the doors (Monty Hall) opens one of the other two doors, revealing a goat, and asks you if you wish to change your selection to the third door (i.e., the door which neither you picked nor he opened). The Monty Hall problem is deciding whether you do.

The correct answer is that you do want to switch. If you do not switch, you have the expected 1/3 chance of winning the car, since no matter whether you initially picked the correct door, Monty will show you a door with a goat.

MathWorld on Monty Hall Problem

 

[Number Six]

I think MathWorld is wrong then. If Monty is standing there and you pick #1 and then he opens another door and reveals a goat and asks you if you want to change your choice, it only makes sense to change if you assume Monty was going to ask you if you wanted to change no matter which door you chose initially.

OTOH, if Monty is standing there and says to you "I know what is behind the three doors and after you make your choice then I'm going to open another door and reveal a goat and ask you if you want to change your choice" then it does make sense for you to switch.

The description on MathWorld sounds like the first case instead of the second.

 

[BillHoyt]

I think MathWorld is wrong then. If Monty is standing there and you pick #1 and then he opens another door and reveals a goat and asks you if you want to change your choice, it only makes sense to change if you assume Monty was going to ask you if you wanted to change no matter which door you chose initially.

OTOH, if Monty is standing there and says to you "I know what is behind the three doors and after you make your choice then I'm going to open another door and reveal a goat and ask you if you want to change your choice" then it does make sense for you to switch.

The description on MathWorld sounds like the first case instead of the second.

"since no matter whether you initially picked the correct door, Monty will show you a door with a goat."

 

[Yaotl]

I think MathWorld is wrong then. If Monty is standing there and you pick #1 and then he opens another door and reveals a goat and asks you if you want to change your choice, it only makes sense to change if you assume Monty was going to ask you if you wanted to change no matter which door you chose initially.

OTOH, if Monty is standing there and says to you "I know what is behind the three doors and after you make your choice then I'm going to open another door and reveal a goat and ask you if you want to change your choice" then it does make sense for you to switch.

The description on MathWorld sounds like the first case instead of the second.

I don't see a practical difference in your two scenarios.

 

[BillHoyt]

I don't see a practical difference in your two scenarios.

If Monty only offered the switch choice when you had originally picked the correct door...