Monty Hall Problem

[Robin]

I'm afraid I don't see the distinction.

The point I'm trying to make is that the host's intentions always matter.

If he always opens a door, then his intentions matter, simply because his intention is to always open a door.

If he sometimes opens a door, then his intentions matter, simply because his intention is to sometimes open a door.

Any way you look at it, his intentions matter.

Oh come on, now you are being silly. The word 'intention' implies a choice, it makes no sense to say someone has an intention if they don't have a choice. When we say he did not have a choice we are saying that it is a premise of the problem that he always offers a choice.

Are you seriously - I mean seriously - saying you cannot see the distinction?

 

[TeaBag420]

I'm afraid I don't see the distinction.

The point I'm trying to make is that the host's intentions always matter.

If he always opens a door, then his intentions matter, simply because his intention is to always open a door.

If he sometimes opens a door, then his intentions matter, simply because his intention is to sometimes open a door.

Any way you look at it, his intentions matter.

NO. And this is the problem I have with conflating the idealized shell game and the Monty Hall problem. The OP stated that Monty opens a door--on one trial. The problem doesn't really begin until he opens the door.

If you don't know his intentions there is no reason to try to take them into account. If you DO know them, THEN it ceases to be a probability problem.

"If he always opens a door, then his intentions matter, simply because his intention is to always open a door."

Does he do the show unintentionally, or does that matter?

 

[Robin]

I have rephrased your example:

<pre>
1. x=4
2. y=3
3. z=Idon'tknow(x,y)
</pre>

Equally helpful, don't you think? Your argument seems to boil down to "I don't know." But in a bad way.

What happened did they declare "miss the point day" and not tell me? This is exactly what I mean't.

If you know that the host had to offer the second choice then it is a probability problem.

If you know that the host didn't have to offer the second choice then it is not a probability problem but may be some other kind of maths problem.

If you don't know whether the host had a choice or not then it is not a problem at all, it is just some incomplete bit of text.

My position all along has been "we need to precisely define the problem before we can find and answer". I did not think that it was particularly controversial.

By the way, is there a "good way" and a "bad way" not to know something? I thought you either knew something or didn't.

If it's not a math problem, maybe it should be moved to another board. Why don't you liaise with the moderators on that?

Did I start this thread? News to me.

I think that you really need to come to the point. Please state what you think the solution to the problem is.

 

[Cabbage]

Oh come on, now you are being silly. The word 'intention' implies a choice, it makes no sense to say someone has an intention if they don't have a choice. When we say he did not have a choice we are saying that it is a premise of the problem that he always offers a choice.

Are you seriously - I mean seriously - saying you cannot see the distinction?

Yes, that's exactly what I'm saying--I don't see the distinction.

How is "always offering a switch" not a choice?

Perhaps you've misunderstood. When I've referred to the host's intentions, or choice, I'm not referring to whether or not he intends, or chooses, to reveal the door in a particular game. I'm referring to his choice on how to run the game in the first place.

It is most certainly a choice on how to run the game:

Let's go back to the original problem, focusing on a game show. How was this game show formulated in the first place? I can imagine the host to be sitting around drinking coffee. He thinks to himself, "Here's the deal, the contestant will try to pick the door containing the prize. Only one of the three doors will contain the prize!"

Then he thinkgs, "Wait a minutel, let's put a little more action into it. Let's say that once the contestant makes the selection, I may reveal one of the doors, and offer the contestant a chance to switch then.

"OK, I might be able to go somewhere with that. Do I want to always offer the switch, or just sometimes?"

He thinks for a moment, and says, "I think I'll do it all the time!"

How can you possibly say that is not a choice on how to run the game?

 

[Cabbage]

First, I don't think it matters how many games are played, so let's agree to throw the stipulation out.

Second, and I'm starting to get this, if the grifter has the option of only offering the switch when you've got the right shell, then in those cases switching gives you a zero chance of winning. If the grifter isn't required to follow rules that you, the bettor know, then this isn't solvable for any arbitrary number of games, which is what I suspect is what you were getting at. Unless you consider it a solution to say "Don't take the bet". Which gets us back to it being a con, and was my original solution.

Because if the grifter doesn't follow known rules, why would someone offer you the bet? And if he doesn't follow known rules, why offer you the bet, absent collusion?

It's a con.

Robin's assertion that the shell game is the same as the Monty Hall problem appears defective under this analysis and I will let you take that up with him.

It's not a con; that was a clearly stated condition of my problem.

Why would the stranger offer the side bet? He may want to offer it because he has faith that some one offering such a game is probably going to be good at it. The shell game man may be expected to be good at running the game, offering the switch sometimes, not offering at others, and doing it in such a way to mislead people into making the wrong decision. He may also be good at reading people, knowing by their behavior what to expect or not when they are present with a switch. Of course, none of this is to say that he cheats--Again, it is most certainly a fair game, the man running it may just be damn good with his strategy of offering a switch at some times, and not offering it at others.

One particular comment you made:

If the grifter isn't required to follow rules that you, the bettor know, then this isn't solvable for any arbitrary number of games, which is what I suspect is what you were getting at. Unless you consider it a solution to say "Don't take the bet". Which gets us back to it being a con, and was my original solution.

Again, it's not a con.

But yes, this is exactly what I'm getting at--The shell game man's strategy is unknown to you.

I wasn't looking for a solution with respect to either taking the bet or not taking the bet; I have been trying to use this example to illustrate that the issue of how the host plays the game makes a huge difference in the odds.

Finally, I don't see how this shell game is any different from the original Monty Hall problem. In the original problem, you are only given information about a single game. Not enough information is given regarding the host's strategy--He opened a door for you, certainly, but maybe that was just your particular game.

 

[Robin]

Cabbage
How can you possibly say that is not a choice on how to run the game?

Is there anybody else here that thinks that the choice on how to run the game n the first place is relevant? Can we assume that the rules don't change during the game?

 

[Cabbage]

Is there anybody else here that thinks that the choice on how to run the game n the first place is relevant? Can we assume that the rules don't change during the game?

Well, a simpler word for "choice on how to run the game" would be "strategy".

It all boils down to the host's strategy.

Maybe he has a complex strategy, offering the switch some times, not offering it at others, all in an attempt to deceive and mislead people into making the wrong decisions.

Or maybe he has a very simple strategy, in which he doesn't even bother to be tricky--he just always opens a door and offers a chance to switch.

The information in the original problem only pertains to the single game in which you are playing. Hardly enough to conclude anything about the host's strategy. But I think it should be clear that the strategies of both you and the host are vitally important in the game, and will affect the probabilities.

 

[Robin]

But yes, this is exactly what I'm getting at--The shell game man's strategy is unknown to you.

Can we make a distinction between strategy and rules.

Teabag is saying that the shell game man will break the rules and palm the pea. But we must assume for the problem that he will not - we must assume that he follows all rules.

Now his strategy is unknown to us - that is OK. But we would not get into any game without the rules being explained to us.

So say we have ruleset 1.

1. There are three cups
2. Shell man puts a pea under the cup
3. The cups are shuffled so that you don't know where the pea is
4. You put your finger on the cup where you think the pea is
5. The shell man lifts one of the other cups and reveals no pea
6. He asks us if we would like to switch
7. Depending on whether you accept the new choice you leave your finger on the cup or move it to the last remaining cup
8. You lift the cup your finger is on and find a pea then you win.

In this case it advantages you to change your choice as long as he follows the rules but the shell man's strategy is irrelevant.

So say we have ruleset 2

1. There are three cups
2. Shell man puts a pea under the cup
3. The cups are shuffled so that you don't know where the pea is
4. You put your finger on the cup where you think the pea is
5. The shell man may lift one of the other cups and reveals no pea
6. If he has lifted the cup in step 5 asks us if we would like to switch
7. If you have been offered a choice you may decide to move your finger to the last remaining cup or leave it where it is. If you have not been offered a choice then you leave the finger where it is.
8. You lift the cup your finger is on and find a pea then you win.

In this case it depends on the shell man's strategy.

 

[Robin]

Well, a simpler word for "choice on how to run the game" would be "strategy".

It all boils down to the host's strategy.

Maybe he has a complex strategy, offering the switch some times, not offering it at others, all in an attempt to deceive and mislead people into making the wrong decisions.

Or maybe he has a very simple strategy, in which he doesn't even bother to be tricky--he just always opens a door and offers a chance to switch.

The information in the original problem only pertains to the single game in which you are playing. Hardly enough to conclude anything about the host's strategy. But I think it should be clear that the strategies of both you and the host are vitally important in the game, and will affect the probabilities.

See my previous post, there are strategies and rules. If the rules state that he must reveal a goat and offer a choice then his strategy is irrelevant.

 

[Cabbage]

Can we make a distinction between strategy and rules.

Teabag is saying that the shell game man will break the rules and palm the pea. But we must assume for the problem that he will not - we must assume that he follows all rules.

Now his strategy is unknown to us - that is OK. But we would not get into any game without the rules being explained to us.

So say we have ruleset 1.

1. There are three cups
2. Shell man puts a pea under the cup
3. The cups are shuffled so that you don't know where the pea is
4. You put your finger on the cup where you think the pea is
5. The shell man lifts one of the other cups and reveals no pea
6. He asks us if we would like to switch
7. Depending on whether you accept the new choice you leave your finger on the cup or move it to the last remaining cup
8. You lift the cup your finger is on and find a pea then you win.

In this case it advantages you to change your choice as long as he follows the rules but the shell man's strategy is irrelevant.

So say we have ruleset 2

1. There are three cups
2. Shell man puts a pea under the cup
3. The cups are shuffled so that you don't know where the pea is
4. You put your finger on the cup where you think the pea is
5. The shell man may lift one of the other cups and reveals no pea
6. If he has lifted the cup in step 5 asks us if we would like to switch
7. If you have been offered a choice you may decide to move your finger to the last remaining cup or leave it where it is. If you have not been offered a choice then you leave the finger where it is.
8. You lift the cup your finger is on and find a pea then you win.

In this case it depends on the shell man's strategy.

I agree 100%.

 

[Robin]

I agree 100%.

But do you agree that with ruleset 1 the shell man's strategy does not matter?

 

[Cabbage]

But do you agree that with ruleset 1 the shell man's strategy does not matter?

I'm not sure what you mean. "Does not matter" with respect to what?

 

[Cabbage]

Sorry, I think I see what you mean now. No it doesn't matter, in the sense that I don't see how the rules allow for any choice or strategy on the host's part.

 

[Robin]

I'm not sure what you mean. "Does not matter" with respect to what?

OK I suspect you have gone into troll mode.

Under ruleset 1 in my earlier post, the shell man's strategy does not affect the outcome of the game as long as he follows the rules as laid out.

Do you agree.

 

[Robin]

Sorry, I think I see what you mean now. No it doesn't matter, in the sense that I don't see how the rules allow for any choice or strategy on the host's part.

OK, so what is your problem with my earlier post in saying that the hosts intentions (or strategy) may not affect the outcome of the game because he may be obliged to offer the second guess under the rules.

 

[Cabbage]

OK, so what is your problem with my earlier post in saying that the hosts intentions (or strategy) may not affect the outcome of the game because he may be obliged to offer the second guess under the rules.

Maybe we misunderstand each other? If there was, and the misunderstanding was my fault, I apologize.

If he is obliged to offer the second guess, I agree, his strategy will not affect the outcome of the game.

However, let's look back at the original problem:

Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors?

Now, at the point the host reveals another door and offers a switch, how do we know whether we are operating under your ruleset 1 or ruleset 2? It is not given in the problem, and cannot be known without additional information (or assumptions).

 

[CurtC]

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.

I'll take one more shot at explaining it to the gentler, kinder TeaBag420. Let's consider these two opposing preludes to the problem statement. In situation A, Monty is an evil guy and will offer a guest the chance to switch his guess only if the guest made the correct guess initially. In situation B, Monty is a brain-dead robot and always reveals a non-winning door after the initial guess, giving the guest the chance to switch to the remaining door.

Now consider the problem statement, as it's given in the OP, following each of these situation statements. You now find yourself in the situation where you've made your initial guess and Monty has offered you the choice to switch. What do you do?

I think it's pretty clear that if he's operating under situation A, you will always lose by switching, and if he's operating by situation B, you'll win more often by switching. Both of these scenarios are completely compatible with what's described in the OP, and we have no idea which is the right one, but the best choice for you to make completely depends on which one it is. Do you want to assume one or the other? Why? As Dirty Harry said, "Feelin' lucky, punk?"

 

[TeaBag420]

Can we make a distinction between strategy and rules.

Teabag is saying that the shell game man will break the rules and palm the pea. But we must assume for the problem that he will not - we must assume that he follows all rules.

GODDAMMIT YOU MOTHERLOVIN' CHANUKAH SUCKER! NO I AM NOT SAYING THAT. I HAVE ACCEPTED CABBAGE'S "CLARIFICATION" (BECAUSE IT IS HIS NEW PUZZLE) WHICH FORBIDS THE GRIFTER FROM PALMING OR OTHERWISE REMOVING THE PEA.

TRY TO FIRETRUCKING READ.

I know you wrote some other stuff, but I didn't read it because you effed up so big, which is not to say I won't respond to it later.

 

[TeaBag420]

What happened did they declare "miss the point day" and not tell me? This is exactly what I mean't.

If you know that the host had to offer the second choice then it is a probability problem.

If you know that the host didn't have to offer the second choice then it is not a probability problem but may be some other kind of maths problem.

So you offer two possibilities, both of which say it's a math problem, but previously you said it's not a math problem. Care to jack, um, I mean explain?

 

[TeaBag420]

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.

I'll take one more shot at explaining it to the gentler, kinder TeaBag420. Let's consider these two opposing preludes to the problem statement. In situation A, Monty is an evil guy and will offer a guest the chance to switch his guess only if the guest made the correct guess initially. In situation B, Monty is a brain-dead robot and always reveals a non-winning door after the initial guess, giving the guest the chance to switch to the remaining door.

Now consider the problem statement, as it's given in the OP, following each of these situation statements. You now find yourself in the situation where you've made your initial guess and Monty has offered you the choice to switch. What do you do?

I think it's pretty clear that if he's operating under situation A, you will always lose by switching, and if he's operating by situation B, you'll win more often by switching. Both of these scenarios are completely compatible with what's described in the OP, and we have no idea which is the right one, but the best choice for you to make completely depends on which one it is. Do you want to assume one or the other? Why? As Dirty Harry said, "Feelin' lucky, punk?"

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.