Monty Hall Problem

[JamesM]

Here's the official Let's Make a Deal website. In the Show Info section, there's a description of the final game:

Sometimes when a Trader had decided to “take the Curtain,” Monty offered to buy it back again… $1,000… $2,000… $3,000 not to take the Curtain! Traders never knew how high he would go.

There's no mention here of Monty revealing one of the other curtains or offering a swap.

There is also a section on The Monty Hall Problem, which reprints the letter that Hall allegedly sent to Steve Selvin. Again, Monty states:

And if you ever get on my show, the rules hold fast for you -- no trading boxes after the selection.

Bill, do you have anything to say on this? I will admit that this is not necessarily much more reliable than wikipedia...

 

[BillHoyt]

Here's the official Let's Make a Deal website. In the Show Info section, there's a description of the final game:

There's no mention here of Monty revealing one of the other curtains or offering a swap.

There is also a section on The Monty Hall Problem, which reprints the letter that Hall allegedly sent to Steve Selvin. Again, Monty states:

Bill, do you have anything to say on this? I will admit that this is not necessarily much more reliable than wikipedia...

Uh-huh. You need to read this article . It is a copy of a New York Times article on the subject:

"Mr. Hall said he was not surprised at the experts' insistence that the probability was 1 out of 2. "That's the same assumption contestants would make on the show after I showed them there was nothing behind one door," he said.

"They'd think the odds on their door had now gone up to 1 in 2, so they hated to give up the door no matter how much money I offered. By opening that door we were applying pressure. We called it the Henry James treatment. It was 'The Turn of the Screw.' "

Mr. Hall said he realized the contestants were wrong, because the odds on Door 1 were still only 1 in 3 even after he opened another door. Since the only other place the car could be was behind Door 2, the odds on that door must now be 2 in 3."

Further down the page, the author effectively puts Hall's quip to Selvin in context:

"according to the rules of the show, ... he did have the option of not offering the switch, and he usually did not offer it."
Hall did not have to allow the contestant to switch. This is what he meant in his letter to Selvin; that Selvin would not be offered the switch.

 

[JamesM]

Thanks Bill, much appreciated!

 

[JamesM]

However, I note in the article it says he "usually did not offer" the switch. How does this square with "he always, always, always, always opened up a klunker door" - would he have opened the door if he wasn't offering a switch? What was the point?

And isn't the Wikipedia entry still correct? The Monty Hall problem doesn't apply to Let's Make a Deal, because the article says Monty was not compelled to offer a switch, and in fact he usually didn't.

Where am I going wrong?

edit: I was of course, completely wrong about why the MHP doesn't apply to Let's Make a Deal: I thought it was because Monty never offered a swap, when in fact it was because he only sometimes offered the swap.

 

[BillHoyt]

However, I note in the article it says he "usually did not offer" the switch. How does this square with "he always, always, always, always opened up a klunker door" - would he have opened the door if he wasn't offering a switch? What was the point?

And isn't the Wikipedia entry still correct? The Monty Hall problem doesn't apply to Let's Make a Deal, because the article says Monty was not compelled to offer a switch, and in fact he usually didn't.

Where am I going wrong?

edit: I was of course, completely wrong about why the MHP doesn't apply to Let's Make a Deal: I thought it was because Monty never offered a swap, when in fact it was because he only sometimes offered the swap.

I wrote that "he always ... opened up a klunker door in the reveal part of the game." What I meant by "reveal part" is the part of the game where he revealed a door as a prelude to offering the switch. (Read the article a bit more; it discusses the psychological gamesmanship in LMAD.)

Go back to the Wikipedia entry to see that I've already edited it again. Cute, eh? The last edit I saw did not say "not compelled to offer a switch," but "not allowed to offer a switch." Wikipedia is a hardly a reliable source of information.

 

[gnome]

gnome,

There are three doors on the stage. Only one of them has the real prize. The probability you got it wrong the first time is, therefore, 2/3.

Sorry. Was thinking backwards. My bad. You're correct.

 

[JamesM]

I wrote that "he always ... opened up a klunker door in the reveal part of the game." What I meant by "reveal part" is the part of the game where he revealed a door as a prelude to offering the switch.

Just to make sure I've got this clear:

He didn't always offer a switch.

When he offered a switch, he always opened another door.

That door was NEVER the door that had the 'real' prize in.

Correct?

Go back to the Wikipedia entry to see that I've already edited it again. Cute, eh?

Hmm, looks like it's been edited back...

The last edit I saw did not say "not compelled to offer a switch," but "not allowed to offer a switch."

You are quite correct, the MHP entry was in error (the author appears to have made the same erroneous assumption I did). I was actually thinking of the Let's Make a Deal entry - it still seems to me that, as it states, the MHP doesn't apply to the real game, except when Monty offered the swap. And even then, we would have to assume that, whenever Monty Hall offered the swap, he did so regardless of what the contestant had chosen. Given the psychological aspects that Monty Hall mentions in the article you linked to, that doesn't seem necessarily obvious to me.

 

[Robin]

Easy. If Monty only offers the switch when you have chosen the car, then you never win by switching.

This is a legitimate criticism. The probability calculations only work if Monty offers the choice indiscriminately.

As I said yesterday, there are two interpretations of the problem. If the second choice is part of the game then Monty's intentions have no bearing on the result.

If the second choice is not part of the game then there is no mathematical answer to the problem.

 

[hgc]

As I said yesterday, there are two interpretations of the problem. If the second choice is part of the game then Monty's intentions have no bearing on the result.

If the second choice is not part of the game then there is no mathematical answer to the problem.

The 2nd interpretation is no interpretation at all, at least not to the problem I presented. The scenario is quite clear: Monty reveals a door and it is a goat. He then gives the option to switch.

 

[Robin]

The 2nd interpretation is no interpretation at all, at least not to the problem I presented. The scenario is quite clear: Monty reveals a door and it is a goat. He then gives the option to switch.

It is not clear it is ambiguous. What you don't say is whether this step is part of the game or whether it is something that the host decided to do off the bat.

That makes all the difference. If this step is part of the game then the answer is "yes - you double your chances by switching your choice"

If this step is optional then there is no mathematical answer.

 

[Yaotl]

It is not clear it is ambiguous. What you don't say is whether this step is part of the game or whether it is something that the host decided to do off the bat.

That makes all the difference. If this step is part of the game then the answer is "yes - you double your chances by switching your choice"

If this step is optional then there is no mathematical answer.

The Monty Hall Problem is as stated though, not as it was on the show. You get the second choice after he reveals a goat.

 

[Robin]

The Monty Hall Problem is as stated though, not as it was on the show. You get the second choice after he reveals a goat.

That is not what I am asking - I can see that from the statement. What I am asking is what is not made clear -

is the step to reveal the goat and offer the switch always done or is it only done sometimes?

If it is always done then there is a specific answer to the question.

If this step is only done sometimes there is no answer to the question.

The actual game show is irrelevant, I am asking about the problem as stated.

 

[Yaotl]

That is not what I am asking - I can see that from the statement. What I am asking is what is not made clear -

is the step to reveal the goat and offer the switch always done or is it only done sometimes?

If it is always done then there is a specific answer to the question.

If this step is only done sometimes there is no answer to the question.

Yes, it is always done. I don't see the ambiguity in the problem. He does open a goat door and you are offered the second choice. There is no may or may not in the actual problem.

 

[hgc]

That is not what I am asking - I can see that from the statement. What I am asking is what is not made clear -

is the step to reveal the goat and offer the switch always done or is it only done sometimes?

If it is always done then there is a specific answer to the question.

If this step is only done sometimes there is no answer to the question.

Always and sometimes are not relevant. I gave a single scenario and described exactly what happened (except for specifying door numbers). No need to extrapolate how it's going to be different the next time.

If the thought experiment or real experiment of a large sample of occurences is useful, then I suggest it be the exact same every time: Monty reveals a door with a goat and offers a switch.

 

[Robin]

Always and sometimes are not relevant. I gave a single scenario and described exactly what happened (except for specifying door numbers). No need to extrapolate how it's going to be different the next time.

If the thought experiment or real experiment of a large sample of occurences is useful, then I suggest it be the exact same every time: Monty reveals a door with a goat and offers a switch.

Still two different problems:

1. If you are describing a single scenario with no information about the conduct of the game then there is no answer to the question. Not enough information has been provided.

2.If it is exactly the same every time then the answer is , "yes you double your chances by switching"

So it really has been settled, all that was required was for the problem to be made more precise.

 

[hgc]

Still two different problems:

1. If you are describing a single scenario with no information about the conduct of the game then there is no answer to the question. Not enough information has been provided.

2.If it is exactly the same every time then the answer is , "yes you double your chances by switching"

So it really has been settled, all that was required was for the problem to be made more precise.

I have no idea what information is missing, or why saying it's multiple occurrences has any effect (remember, I'm asking about probability, not about results). Please tell me how you would phrase the problem to clear up the ambiguity.

 

[Drooper]

I think one of the reasons why this problem always causes so much controversy is because it is being analysed in the wrong way.

Most people tend to use probability theory alone, but this can only be done in this case with full information about the participants (as many posters are pointing out - e.g. is Monty trying to double bluff you?)

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).

 

[Robin]

Yes, it is always done. I don't see the ambiguity in the problem. He does open a goat door and you are offered the second choice. There is no may or may not in the actual problem.

This is where the misunderstanding has occurred. Nobody spotted the ambiguity in the question.

If the question described the way the game is played then it is reasonable to assume that this step is always carried out. If on the other hand the question describes just one single scenario then we have no way of knowing if any particular step is required or if it has just been done on this occasion.

Some people read it one way, some people the other. No one was right or wrong but we should proceed on the basis that there are now two versions of the problem.

 

[hgc]

I think one of the reasons why this problem always causes so much controversy is because it is being analysed in the wrong way.

Most people tend to use probability theory alone, but this can only be done in this case with full information about the participants (as many posters are pointing out - e.g. is Monty trying to double bluff you?)

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).

Not necessary. This is a pure, sterile logic problem. The reference to "Monty Hall," "doors," "car," "goats," and other particulars is only to give it flavor.

 

[Robin]

I think one of the reasons why this problem always causes so much controversy is because it is being analysed in the wrong way.

Most people tend to use probability theory alone, but this can only be done in this case with full information about the participants (as many posters are pointing out - e.g. is Monty trying to double bluff you?)

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).

You can use probability if the offer to switch is a consistent part of the game. In this case it makes no difference if Monty is trying to bluff you.

If the offer to switch is not always done then you can't use probability. I am unable to say whether game theory would be useful.