Monty Hall Problem

[TrueSceptic]

Yes... and there is the 3rd option... Monty is only offering the choice because you have the car. But if you don't know which of these options are in play your odds are the same as in the classic situation. See post #252.

I assumed that that was ruled out, but as you say, if you don't know if it has, then you are right.

Of course, a capricious Monty might want to you to win because he likes you, or he's just, well, capricious.

In the classic situation the host cannot be blind, because you are always shown a goat and the host cannot be in the 3rd category, because it's stipulated that you are always offered a choice. So you are either in the classic situation or you don't know which of the 3 situations you are in. This means that IF you are offered the choice (after being shown a goat)-- you increase your odds by taking it. You won't always win-- but the odds are double what they will be if you never take the choice. (Your actual odds in the "unknown" situation are "probability that the car is behind the door you pick" multiplied by the "probability that you will be given a choice"-- but it ends up being the same odds as in the classic problem.)

You've obviously played this through many times (with your class?). I'm glad to see this discussion because this problem has a habit of taking over my mind in unguarded moments and it's sort of nice to see those quandaries expressed so well here

I take it that you know the similar Three Prisoners Problem?

 

[GreyICE]

Just for the record, my opinion of you is on par with your opinion of me. I find you self-important, rambling, and strident; moreover, I find it hypocritical that you claim I don't admit I'm wrong. I do and have. I've never witnessed you doing so, however. Nor Claus.

Well given the fact that you have me on ignore 4/5ths of the time, at least according to you, I'm really not surprised. I'm shocked you saw that post.

Lets see, what's the most passive-aggressive response you could make to this? Back to the ignore list with you?

However, if I've made a wrong statement here-- no-one has highlighted it nor shown a link or reasoning as to why I am wrong. Feel free to do so.

Nobody cares anymore. Really.

I was trying to explain the problem as I do to my students, when Claus derailed insisting I was wrong about something and that I didn't know anything about the problem. I think any perusal of my explanations by anyone who really understands the problem will affirm that my answers were correct--and that I've always maintained the following:

In the classic Monty Hall Problem (or in situations where you don't have knowledge as to whether it's the classic Monty Hall Problem but it could be-- because you are shown a goat and offered a choice)-- you have a 1/3 chance of winning by keeping the first choice and 2/3 of a chance of winning by switching to the other door.

If you have a problem with that statement, highlight the problem and prove me wrong. Otherwise, I'll take it you were wrong, like Claus, and are mad at me for revealing your pompous buffoonery.

My pompous buffoonery. Well, now I know how you think I don't back down. You fail to read threads.

If you're shown 3 doors, pick one, the host opens one, and its a goat, you don't have enough information to determine whether its a good choice to switch.

The reason this thread is so rambling is because Claus derailed to make allegations about me-- just as you have done. I can let your dishonest allegations stand, or I can defend myself revealing you guys for what you are. This time I chose the latter, because some people actually wanted to understand the problem. But there won't be a next time, because I won't waste my time reading your posts.

Oh wait! I didn't even have to wait for the next post for the passive-aggressive back to the ignore list.

Do we have a facepalm smiley?

I did reply to this so I didn't pull an 'oracle of the ages' and write some pithy response and zinger out of here, but you continue to exemplify exactly what I was talking about. Look, you and Claus disagree. We know you disagree, we know you'll fight about it for 10k pages. Can you just, I dunno, put him on ignore or something?

 

[ddt]

Articulett vs. Claus round... god. What are we on now?

Claus, [...] Maybe it's the fact that you back down even less frequently than I do, and have a bulldozer-like obstinacy combined with an unerring knack for annoying even people who agree with you.

Articulett, you're ornery, have no capacity to admit you're wrong, frequently post long rambling posts that go nowhere, [...]

Therefore, I've got to ask, since we can basically assume that the two of you will disagree just to be disagreeable, can we just manually insert 'three pages of articulett and claus arguing over some nonsense point that adds nothing to the discussion' at the bottom of pretty much every OP and then we don't have to, I don't know, do it?

I'm not long enough around to judge on all you wrote here GreyICE, but I can heartily agree with the above. Especially, articulett, where do you get the time to write all that? I'd need two clones to just read the stuff...

BTW, with all this talking of goats: where is the Marquis?

 

[Herzblut]

Yes... and there is the 3rd option... Monty is only offering the choice because you have the car. But if you don't know which of these options are in play your odds are the same as in the classic situation. See post #252.

This is incomprehensible. In this version, the winning chances of a player changing the door when possible are identically zero.

 

[TrueSceptic]

This is incomprehensible. In this option, the winning chances of a player changing the door when possible are identically zero.

Zero means you know the outcome. We don't even know the rules.

 

[TrueSceptic]

Damn you, Monty Hall!

You trapped me in your endless mind-games again!

 

[Myriad]

Yes, absolutely!




That's why I specifically said "will switching improve my chances of winning?" in the post you quoted, rather than simply "should I switch?"

Because when you boil it right down like that, then the answer is a simple "yes".

That's an excellent point. The correct answer to "should I switch?" depends entirely on whether or not I picked the car the first time!

Respectfully,
Myriad

 

[articulett]

Point of information.

GreyICE admitted a mistake several pages back on this thread, with perfectly good grace.

I'm going away now, because you're only quibbling about semantics.

And I don't think anyone is reading my posts anyway.

Rolfe.

Point of information: GrayICE made a false allegation about my capacity to admit error which you and others have agreed with. Claus has made this same allegation and multiple people and postings were shown proving it wrong. Did you have actual evidence of a single case where I was shown to be wrong and did not admit it?

Moreover, my statement was in response to GreyICE's allegation--and it was stated as an opinion and it is TRUE. I claimed that I had never seen GreyICE admit error. I will change that if you want to cut and paste the event. I will then be able to claim that I have witnessed him admitting error. However, I won't hold my breath that those who've made the accusation against me will apologize for making a false allegation. Nor will they show any interest in the evidence which proves they made a false allegation.

You've accused me of things that you are more guilty of... as has GreyICE and others who agreed with his statement.

I agreed with everything you wrote in your little piece-- but I did not respond because you agreed with a false allegation made about me AND went on to accuse me of making a false allegation when I stated that I had not witnessed (the truth) GreyICE apologizing for an error. All you needed to do was cut and paste so I could "witness" it. I might add that I've never witnessed an admission of error from any of the people who allege that I have no capacity to admit I am wrong or jumped on that bandwagon. To say I have no capacity to admit error is a LIE that is betrayed by multiple posts and witnesses on this forum. Are you going to be apologizing for this error? Why don't you try modeling the kind of person you would prefer I be... because as far as I can tell, the allegations GreyICE made (and you agreed with) are more fitting of you than of me. Why don't you show us all how good you are at admitting this error?

If you think I made an error and didn't admit it, feel free to highlight it. If it is an error-- I guarantee, I'll admit it.

If you can show me that GreyICE has admitted error... then I will be able to change my claim that I have never witnessed it.

As for the Monty Hall Problem... I think we can all see that in the classic case, you always have a 1/3 chance of winning by not switching and 2/3 chance by switching. Those odds don't change so long as you are staying in the classic problem despite Claus' claim (and this derail where allegations have been made about me) that supposedly "the Monty Hall problem is all about the odds changing when the host gives you a choice."

 

[Mobyseven]

Point of information.

GreyICE admitted a mistake several pages back on this thread, with perfectly good grace.

I'm going away now, because you're only quibbling about semantics.

And I don't think anyone is reading my posts anyway.

Rolfe.

Nah, I'm reading. Just nothing really to say other than a hearty, "I agree."

 

[Lisa Simpson]

Please - let's not let this thread slip into personal bickering and the magical land of AAH and infractions. Discuss the topic and not each other.

Replying to this modbox in thread will be off topic  Posted By: Lisa Simpson

 

[jsfisher]

Lisa,

Monty says thanks.

 

[Michael C]

At the risk of repeating myself.... (well, the whole thread is on a repeat cycle I suppose)...

Thanks, Rolfe, for that clear and eloquent post. I agree with you that much of the fascination comes from looking at the different possible scenarios that the ambiguous wording of the problem permits. It seems that some people, having twigged to the fact that in the so-called "classic" version you double your chances of winning by switching, just stop there and refuse to consider other interpretations. I was particularly bugged by the posters on this thread who flatly refused to admit that the answer could change, depending on whether or not Monty knows where the prize is.

[...]But what if Monty is totally capricious? Maybe he'll help some contestants and be a bastard to others. Maybe he has a different scenario in his head every time the game is played! From what I've heard from those who have watched the original show, this may well have been the actual case. And presumably it was a perfectly viable game show.

With my limited knowledge of game shows, I tend to think that the "Monty-is-capricious" strategy is the one most likely to be used in real life. I don't think that the strategy where Monty chooses the door at random is often used. If the host doesn't know what is behind the doors, it's more likely that he lets the contestant choose a door to be opened, with the understanding that whatever is found behind the door is lost to the contestant.

It's also clear that the "Monty-is-a-bastard" technique will be found out if it is used consistently. The "classic" scenario (the host always opens a door himself, there's always a goat behind it and he always gives the contestant the chance of switching), is also too boring if it's used every time. In order to keep up the suspense and make sure that no contestant has a chance of creating a consistent winning strategy, a good game show host has a whole repertoire of variations: not only does he sometimes open a door himself, sometimes let the contestant do it, sometimes let the contestant switch and sometimes not: he has other tricks such as offering the contestant a cash sum for whatever might be behind the chosen door.

Finally: there's no doubt for me that the OP does present the problem in an ambiguous way, leaving open the possibility of the "Monty-is-capricious" strategy. For that reason one could consider that the only answer is "there is not enough information to decide", but I think the much better answer is to expose the differing odds that result from Monty's different possible strategies. Which is what you've done.

 

[sphenisc]

..But what if Monty is totally capricious?...

Yee gads! Could you have chosen a more ambiguous turn-of-phrase??!

 

[Mashuna]

Ok, I think I've followed this so far. But what if you open a door to find it completely empty, then turn around to see a car heading for the exit driven by a goat?

 

[Rolfe]

Yee gads! Could you have chosen a more ambiguous turn-of-phrase??!

Oh wow! My puns are so good I don't even spot them myself until someone else points them out!

By the way, can anyone tell me what year this Monty Hall show started?

Rolfe.

 

[TrueSceptic]

Oh wow! My puns are so good I don't even spot them myself until someone else points them out!

By the way, can anyone tell me what year this Monty Hall show started?

Rolfe.

From Wiki,

Hall was the host of the long-running game show Let's Make a Deal, which he developed and produced with partner Stefan Hatos. Let's Make a Deal aired on NBC daytime from December 30, 1963 to December 27, 1968 and on ABC daytime from December 30, 1968 to July 9, 1976, along with two primetime runs. It also aired in syndication from 1971 to 1977, from 1980 to 1981, from 1984 to 1986, and again on NBC briefly from 1990 to 1991.

Now to read the rest of this improbable thread.

 

[Ivor the Engineer]

<snip>

However, I'm not playing with computers. There is an actual person there. Monty just opened a door to reveal a goat. Will switching improve my chances of winning? Well, the answer depends entirely on whether or not Monty himself knows where the car is.

Surreal, what?

Rolfe.

The answer depends on what you know or assume about the host's behaviour and the result of one of the not-chosen doors being opened.

I agree with your argument that the only reasonable behaviour of the host for a gameshow would be to always open a not-chosen door. Under this condition, if the host is guessing or knowledgeable but malevolent, the show might finish early with the host revealing the car.

 

[Rolfe]

The answer depends on what you know or assume about the host's behaviour and the result of one of the not-chosen doors being opened.

Actually, since the behaviour hasn't happened yet, it depends on what we assume his intentions are.

Rolfe.

 

[Rolfe]

I agree with your argument that the only reasonable behaviour of the host for a gameshow would be to always open a not-chosen door. Under this condition, if the host is guessing or knowledgeable but malevolent, the show might finish early with the host revealing the car.

That wasn't exactly what I said. I think the most likely behaviour for a game show host in real life is to be capricious (pun now intentional!).

Consistent "bastard" behaviour would be very poor entertainment as it would soon become obvious that nobody who switched ever won the car, indeed that the very offer of the switch was a guarantee that you'd chosen right the first time.

The other two consistent behaviour patterns (always opening an unchosen door to reveal a goat, and always opening an unchosen door at random) would be perfectly viable game show scenarios, but because they would present less variety than capriciousness, they might well be less good entertainment value.

However, when the puzzle is presented as a brain-teaser, I think it's unreasonable for the basic premise to be capricious behaviour, because in that case there is simply no answer, making the question pointless. Capricious behaviour means that all three of the possible scenarios may be included, including the "bastard" scenario, in which switching is fatal. Thus, Monty may be running a scenario where switching is beneficial, or one where switching is neutral, or one where switching is fatal. If you can't say which one, then the brain-teaser is just silly.

We may therefore assume consistent behaviour as the premise for a brain-teaser. And we may exclude consistent "bastard" behaviour on the basis that the brain-teaser is about a game show, and consistent "bastard" behaviour is incompatible with a realistic game show.

Thus we're left with the two possible consistent behaviours, either the "classic" one where Monty knows where the car is and will always reveal a goat at step 2, and the alternative which is introduced by the ambiguous wording of the question, in which Monty has no idea where the car is and has opened a door at random.

The interesting one to ponder is the situation where one of these two scenarios is in operation, but you don't know which.

Rolfe.

 

[Ivor the Engineer]

Rolfe,

The only behaviours Monty could be employing are:

1) "Bastard": Reveal the car if the contestant had not chosen the door with it behind, else reveal a goat.

2) "Random": Monty doesn't know which door has the car behind it and opens a door at random, which could be a goat or the car.

3) "Kind": Monty knows which door has the car behind it and if it is one of the unchosen doors, always opens the door with the goat behind it.


Behaviour 1 results in contestants either never getting the opportunity to switch doors, since Monty reveals the car, or switching doors results in them loosing the car.

Behaviour 2 results in contestants either never getting the opportunity to switch doors, since Monty reveals the car, or switching giving contestants a 2/3rd chance of winning the car.

Behaviour 3 results in contestants winning the car 2/3rds the time if they switch doors.

Now, if it is made clear that contestants always gets the opportunity to switch doors (and it always has been in the versions of the problem I've seen), then behaviours 1 and 2 are not possible.

Alternatively, if sometimes Monty is allowed to reveal the car after the contestant's first choice, either because he doesn't know which door it is behind (behaviour 2), or he doesn't want the contestant to win (behaviour 1), then capriciousness is possible and it becomes impossible to answer the question, as you have already said.

Here's another similar puzzle:

A random two-child family whose older child is a boy is chosen. What is the probability that the younger child is a girl?

A random two-child family with at least one boy is chosen. What is the probability that it has a girl?

The answers and explanation are on wiki, under 'Boy or Girl paradox'.