Monty Hall Problem

[Yaotl]

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.

No, it's quite clear. There is no real game here, just this one instance. You are given all the facts that you need to come up with the answer.

 

[hgc]

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.

Once again I ask: what's always got to do with it? I am asking about a single scenario, where all the relevant data (in my opinion) is given, and a probability of 2 outcomes is requested.

 

[Robin]

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.

OK here are the two versions:

Version 1 - The game is conducted as follows:
1. There are three doors
2. Behind one door is a car
3. Behind the other two doors are goats
4. You guess which door contains the car but the door is not opened yet
5. The host opens one of the other doors revealing a goat and asks if you would like to make a new choice
If you were a contestant on the above game would making a new choice increase your chances of picking the car?

Version 2 - The game is conducted as follows:
1. There are three doors
2. Behind one door is a car
3. Behind the other two doors are goats
4. You guess which door contains the car but the door is not opened yet
5. The host will either open one of the other doors revealing a goat and ask you if you want to switch, or he may just go with your first choice
If you were a contestant on the above game would making a new choice increase your chances of picking the car?

Two different version, two different answers.

 

[hgc]

OK here are the two versions:





Two different version, two different answers.

Version 1 is the question I posed. Version 2 is not. There is no way Version 2 can be construed as my scenario. Like I said before. Don't imagine other scenarios. The question provides and requires exactly one scenario, and your Version 1 is that scenario.

 

[Drooper]

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.

I believe game theory is the correct way to analyse the problem in its general form, meaning the choice made (to reveal or not reveal) by the Monty Hall is not known in advance. It deals with such choices under imperfect information.

 

[Robin]

Once again I ask: what's always got to do with it? I am asking about a single scenario, where all the relevant data (in my opinion) is given, and a probability of 2 outcomes is requested.

As explained in my earlier post 'always' has everything to do with it. The reason is that there is an additional element of uncertainty and you don't know whether it was random or whether it was dependent on some other event. That makes a probabilistic answer impossible.

 

[Robin]

No, it's quite clear. There is no real game here, just this one instance. You are given all the facts that you need to come up with the answer.

Again you are misunderstanding, I have never considered the real game relevant.

But the question of whether Monty had the option of not offering the second choice is very much relevant. Here is an event that we cannot assign a probability to.

I have set out the two versions above in reply to hgc. If you are speaking of version 1 then the answer is "p=0.333 if you don't switch, p=0.667 if you do"

If you are speaking of version 2 then there is no probabilistic answer.

(edited to fix typo)

 

[hgc]

As explained in my earlier post 'always' has everything to do with it. The reason is that there is an additional element of uncertainty and you don't know whether it was random or whether it was dependent on some other event. That makes a probabilistic answer impossible.

Aye Carumba! The only random event is what is the initial selection, and if that was the car, which of the other 2 doors will be revealed. There are no other random events. What on Earth could effect the outcome that is not accounted for in the original question?

 

[pgwenthold]

Once again I ask: what's always got to do with it? I am asking about a single scenario, where all the relevant data (in my opinion) is given, and a probability of 2 outcomes is requested.

Because you don't know why you have been given the option.

As I noted (and Robin has been saying), if you were only given the option because you guessed correctly, then you would lose when you switched.

This is why always is important (actually, to be fair, it can also be indiscriminately, as opposed to always). Unless you know that Monty gives you the option indiscriminately, then you cannot know that he has not done it only when you have chosen the correct door (or it could be only when you chose the wrong one, for that matter).

Nothing in the original description requires that the option was given indiscriminately. Given that, you do not have enough information to make the decision.

Of course, it seems pretty clear to me that the intent of the problem is that it is supposed to be an assumption of offering the option indiscriminately, but it is certainly not required.

Regardless, the answer is not that your chance of winning is 1/2.

 

[hgc]

Again you are misunderstanding, I have never considered the real game relevant.

But the question of whether Monty had the option of not offering the second choice is very much relevant. Here is an event that we cannot assign a probability to.

I have set out the two versions above in reply to hgc. If you are speaking of version 1 then the answer is "p=0.333 if you don't switch, p=0.667 if you do"

If you are speaking of version 2 then there is no probabilistic answer.

(edited to fix typo)

There is zero question of what Monty will do. I described what he did. There is a single scenario. Everything you need to know is known. What Monty will do next time doesn't matter, because the problem posits no next time.

 

[Robin]

Aye Carumba! The only random event is what is the initial selection, and if that was the car, which of the other 2 doors will be revealed. There are no other random events. What on Earth could effect the outcome that is not accounted for in the original question?

Please think about this for a while before answering. If offering the second choice was a decision that the host made then that decision would affect the outcome. For example if the host only offers a second choice if you chose right there would be a different outcome than if the host offered a second choice if you chose wrong.

If the second choice was not a decision then it does not affect the outcome.

 

[pgwenthold]

Aye Carumba! The only random event is what is the initial selection, and if that was the car, which of the other 2 doors will be revealed. ?

and whether to reveal what is behind the door. Actually, it has to be either random or always. If it is only when you choose the car, then you are going to lose by switching.

However, you cannot tell from only one sample whether it is an indiscriminate thing or whether it was revealed because you picked the car.

 

[Robin]

I believe game theory is the correct way to analyse the problem in its general form, meaning the choice made (to reveal or not reveal) by the Monty Hall is not known in advance. It deals with such choices under imperfect information.

However if Monty did not have a choice, ie he was required to offer the switch as part of the rules of the game then game theory is not required, it becomes a relatively simple probability problem.

 

[Robin]

Version 1 is the question I posed. Version 2 is not. There is no way Version 2 can be construed as my scenario. Like I said before. Don't imagine other scenarios. The question provides and requires exactly one scenario, and your Version 1 is that scenario.

I didn't imagine it, epepke and gnome pointed out this interpretation. I disagreed with them initially but on thinking about it I can see that you could read the question that way - ie there are 3 possible decisions to consider:

1. You initial choice
2. Monty's decision to reveal the goat and offer the switch
3. Your decision whether or not to switch

If version 1 is the question then the answer is as I stated, the probability of picking the car is 0.333 if you dont switch, 0.667 if you do.

 

[pgwenthold]

There is zero question of what Monty will do. I described what he did.

But you do not know why, and knowing why is critical, because:

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

If you have a single instance, and you are shown a door, then you don't know if it is because Monty knows you have chosen a car and goes by (1), Monty knows you have chosen a goat and goes by (2), or this is just the way he always does it, or this is the way it happens sometimes.

 

[Robin]

But you do not know why, and knowing why is critical, because:

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

If you have a single instance, and you are shown a door, then you don't know if it is because Monty knows you have chosen a car and goes by (1), Monty knows you have chosen a goat and goes by (2), or this is just the way he always does it, or this is the way it happens sometimes.

Thank you you put it better than me. The only other option I would add is:

4) Monty was obliged to reveal the goat and offer the switch as part of the rules of the game in which case switching means you will win 2/3 of the time.

 

[rppa]

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.

I've seen the statement that it always pays to switch except in the malicious-host assumption: An offer sto switch only comes when he knows you have the car. Another behaivor, which has been raised here, is that sometimes he offers to switch and sometimes he doesn't.

But of course you don't know whether Monty is feeling malicious or fair today, so somehow you have to make an optimum choice across multiple hypotheses.

 

[Hellbound]

Have to agree that "knowing why" is not a part of the original problem, but an extension of the problem to a generalized case, rather than the specific.

The question stated that the goat was shown. Period.

What you're doing in trying to make scenario two and trying to analyze why is adding in more trials, essentialyl adding a probability to the goat step. As such, you are changing the original problem (it then becomes "Monty sometimes opens a door" not "Monty opens a door"). You are going from a single scenario to a multiple grouping of scenarios.

Instead of thinking of this as a game show, think about it as a single, one-time only, never-to-be-repeated special event (like at a football game or something). He does offer you the choice, you are shown a goat.

The given answer (2/3 chance for switching) is ALWAYS correct for the problem as stated. To include the idea that you only get offered the choice if your door is the car is to assume multiple trials and to violate the statement that you are shown a goat. You have now added an additional problem to the original, and changed the outcomes.

For a single event, analyzed, the answer is correct for the problem as stated. Only if you look at multipl trials, and only iff the open option is not done always, does the probability change. It requires you to go beyond the original problem to develop an alternative answer. You say it requires more information, and that is the point. Getting more information takes the problem outside it's original boundaries and changes it into a new problem.

 

[Johnny Pneumatic]

What were the odds that this thread would have over 150 posts after one day?!

 

[Robin]

I've seen the statement that it always pays to switch except in the malicious-host assumption: An offer sto switch only comes when he knows you have the car. Another behaivor, which has been raised here, is that sometimes he offers to switch and sometimes he doesn't.

But of course you don't know whether Monty is feeling malicious or fair today, so somehow you have to make an optimum choice across multiple hypotheses.

As I said, if the host has a choice at all there is no probabilistic solution.

If the host has no choice but is obliged to offer the second choice then malicious or fair makes no difference - switching will always double your chances of picking the car.