Monty Hall Problem

[Michael C]

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.

I doubt that the host would knowingly open a door with a car, nor run the risk of opening a door with a car by choosing at random. It wouldn't make for a good game show. Imagine the situation if the host says "OK, you've chosen door A: let's see what's behind door B"; he opens door B and there's the car. End of game. People will immediately suspect him of knowing where the car was, even if he didn't.

It's a completely different thing if he lets the contestant choose which door to open: the contestant dithers between doors B and C, people in the audience are shouting out which door they think should be chosen, the contestant finally makes a decision and the door is opened. If the car is found there it's a disappointment, but it was the contestant who sealed his own fate: it looks as if he's been given a fair shot.

So I think a real-life game show host will be switching between knowingly revealing a goat, or letting the contestant decide. Of course, he can also try to push the contestant to make a certain decision...

 

[CurtC]

The exact wording of that publication isn't stated, but going from CurtC's and Claus's insistence that the "Monty will always open one of the two unchosen doors and will always reveal a goat" version is The Only True Authentic Monty Hall Puzzle, I'm assuming that perhaps it did make this clear.

Actually, in the game show, Monty was nothing if not capricious. The very idea of the show was Monty presenting people with various and sometimes bizarre choices. For example, in a situation like the puzzle, he may offer a switch or not, or may offer a wad of cash instead of the chosen door, etc. etc. That said, the Only True Authentic Monty Hall Puzzle is not that - the proper puzzle explicitly says that not only does Monty know where the prize is located, he must first show you a non-prize door and offer the switch.

And once you think about it, that is also a reasonable interpretation of the wording of the puzzle as usually presented. And, indeed, a reasonable way of running a game show.

Thank you - the way the puzzle is usually presented does not properly constrain Monty's rules. In this case, I thought at first it was the same, but I can see some ambiguity where people (especially non-native English speakers) could interpret the wording in the OP to specify that Monty was forced to behave as he did.

However, it's not a perfectly viable brain teaser. To have any validity as an abstract puzzle, there has to be consistency. To put the question to anyone and then say, well, there's no right answer because Monty is entirely capricious, isn't a lot of fun.

Sure, but no one presents it as a brain teaser with the intention of bringing in all the other variations of Monty's behavior. I've seen this come up here and in other forums over the years, and the pattern is this: Someone posts a poorly-worded formulation of the puzzle, answers are offered that the answer is to switch, you'll win 2/3 of the time. Then some curmudgeon like me points out that, according to the puzzle given, you can't say that for sure, and it's actually interesting to consider the different possibilities of the rules Monty may be playing by, and how that affects the probabilities.

Usually, it ends there, but there has been a lot of bickering in this thread revolving around misunderstandings of people's interpretations of wordings, etc.

 

[ZouPrime]

OMGLOL at anyone trying to redefine a problem that has been as much discussed as the Monty Hall puzzle. Seriously. There are so many web pages dedicated to it, any attempt of redefinition is ridiculous.

http://en.wikipedia.org/wiki/Monty_Hall_problem

It's really not hard.

Anyway.

The odds may be 2/3 if you switch, but 0.999... < 1.0 and there's nothing anyone can say to make me change my mind of that!

 

[CurtC]

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.

...

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.

I hope you mis-typed. IF we're considering the case where Monty picks a door at random, and it just so happens that when it's your turn, he revealed the goat, then (in that case), it's a 50/50 proposition - the odds are equal for the two remaining doors.

 

[Ivor the Engineer]

I hope you mis-typed. IF we're considering the case where Monty picks a door at random, and it just so happens that when it's your turn, he revealed the goat, then (in that case), it's a 50/50 proposition - the odds are equal for the two remaining doors.

I was considering the case where the contestant chooses a door and then Monty opens one of the other two doors at random. If Monty does not open the door with the car behind it (doing so would end the show there and then), the contestant would win 2/3rds of the time by switching to the door Monty did not open.

 

[CurtC]

I was considering the case where the contestant chooses a door and then Monty opens one of the other two doors at random. If Monty does not open the door with the car behind it (doing so would end the show there and then), the contestant would win 2/3rds of the time by switching to the door Monty did not open.

Imagine that you're on a game show where the hosts always operates by letting you pick, then he, not knowing where the prize is, opens one of the other two doors at random, then if the prize is still hidden, offers you the choice of switching. Now, at this point, what should you do? Which door has the best chance of winning?

This is the situation we're discussing right now, do you agree? I have to clearly state that up front so that some others here won't complain that this isn't the classic version of the puzzle, it's a digression imagining some different rules.

In this case, each door has equal probability. I think that once someone understands the 2/3 answer for the classic puzzle, this variation is just as interesting. Here's how they add up. Imagine 300 people who over the years start playing the game, and let's count their numbers.

Group 1: picked a goat door on their first guess, and then Monty revealed the car. Game over. There are ~100 people in this group.

Group 2: picked a goat door on their first guess, and then Monty revealed the other goat door. There are ~100 people in this group, and they would all win by switching.

Group 3: picked the car door on their first guess, then Monty (of course) reveals a goat door. There are ~100 people in this group, and they would all lose by switching.

Now, only the people in groups 2 and 3 have found themselves in the situation described. Group 1 does not count towards this analysis, because the problem statement has already ruled them out. Out of those remaining 200 people, 100 would win by switching, 100 would win by sticking.


ETA: Actually, I think that this is a much more interesting concept than the classic "2/3" puzzle. If the puzzle is not worded properly, and leaves out the rules Monty must play by, the correct answer depends on what assumptions you make about why Monty offered the switch.

 

[sol invictus]

I was considering the case where the contestant chooses a door and then Monty opens one of the other two doors at random. If Monty does not open the door with the car behind it (doing so would end the show there and then), the contestant would win 2/3rds of the time by switching to the door Monty did not open.

No! Switching will win 1/2 the time in that situation. That's what the last 5 pages of this thread are about....

 

[CurtC]

I'm not a native speaker, so what is unclear about

The player picks a door, then Monty opens one of the two other doors and reveals what is behind and offers the player to switch to another door of his choice.

Some people here have really gotten their panties in a wad about this very aspect, so I'm trying to be polite a clear...

That statement reads to me like a description of what has already happened. Even though it doesn't use the past tense, it's a common way of speaking informally. On the other hand, I can see how someone else may interpret the present tense as specifying the rules Monty must play by. I haven't done a poll, but my guess is that most native English speakers would, in similar circumstances, interpret the wording as simply reporting what had already happened.

That's what's unclear about it.

 

[ZirconBlue]

A funny thing about humans is that they tend to go with their first choice-- as soon as they place a bet they tend to see their odds as increasing-- they feel more confident with their choice... but your odds don't suddenly change when you place your bet or pick your door... your feeling that you are more likely to win, does not make you more likely to win. This is true even with lottery winners. Before playing, they may correctly assess their odds-- but once they've committed, they "feel" as if their odds go up.

My whole state (Nevada) is built on gambling odds and the bizarre psychology and irrationalities people have in regards to the subject-- their belief in "winning streaks" and so forth.

[bolding added]

I thought I'd point out that articulett's original claim was entirely correct. The act of selecting a door (or placing a bet) does not, in and of itself, change the odds of that door (or bet) being the correct (winning) choice, at that point in time. The act of selecting a door does not make that door special or somehow more likely to be the correct door, but people often behave as if it did, favoring their first choice, even when the situation changes. If the solution to the classic Monty Hall problem were obvious to most people, then it wouldn't be very interesting. That it defies the "common sense" understanding is what makes it worth discussion in the first place, IMO.

 

[Ivor the Engineer]

Sol & CurtC, you are correct (but you already knew that). I've just drawn out the tree diagram and convinced myself my previous intuition was wrong.

The difference is the probability Monty opens a particular door in the original problem is conditional in one case on the door the contestant chooses, whereas in the "dumb" Monty version, he (typically) chooses the door to open with equal probability.

Where it gets interesting is if you now say although Monty is ignorant to which door has the car behind, he still alters the probability of opening a particular door based on the door the contestant chooses.

 

[articulett]

[bolding added]

I thought I'd point out that articulett's original claim was entirely correct. The act of selecting a door (or placing a bet) does not, in and of itself, change the odds of that door (or bet) being the correct (winning) choice, at that point in time. The act of selecting a door does not make that door special or somehow more likely to be the correct door, but people often behave as if it did, favoring their first choice, even when the situation changes. If the solution to the classic Monty Hall problem were obvious to most people, then it wouldn't be very interesting. That it defies the "common sense" understanding is what makes it worth discussion in the first place, IMO.

Thank you.
I am forever in debt to you.
The whole derail was because of that sentence.

*kisses zirconblue all over*

 

[Rolfe]

Sol & CurtC, you are correct (but you already knew that ). I've just drawn out the tree diagram and convinced myself my previous intuition was wrong.

The difference is the probability Monty opens a particular door in the original problem is conditional in one case on the door the contestant chooses, whereas in the "dumb" Monty version, he (typically) chooses the door to open with equal probability.

Where it gets interesting is if you now say although Monty is ignorant to which door has the car behind, he still alters the probability of opening a particular door based on the door the contestant chooses.

Mmmm, yes.

If you've got to the point in the game where you have chosen a door and Monty has then opened another to reveal a goat, the answer to the question "will switching to the other unopened door improve your chances of getting the car" depends entirely on Monty's motive in choosing which door to open.

A. "Classic" scenario, where Monty knows where the car is and has deliberately opened a goat door, switching doubles your odds of winning (from 1/3 to 2/3).

B. Alternative possibility, not excluded by the way the puzzle is usually worded, where Monty does not know where the car is (or if he knows, is capable of entirely disregarding this information) and has opened one of the two unchosen doors at random. Switching does not change the probability of getting the car. (Originally 1/3 chance you chose correctely the first time, 1/3 chance the remaining door has the car, and 1/3 chance that Monty might have revealed the car - we are now at the stage where that last possibility has been excluded, and you're left with two equal-probability coices.)

C. "Monty-is-a-bastard" scenario, where he knows you have already chosen the correct door and is only offering you the switch to entice you away from the winning choice. Switching would lose you the car 100% of the time. However, as this is not a strategy (if employed consistently) that would produce a viable game show, it is arguable that it may be discounted.

The inclusion of the last scenario is mainly of importance when considering the possibility that Monty may be running a different scenario every time (which is probably what actually happened in the real show). While scenario C is non-viable if used consistently, it could form part of the mix in this situation.

Scenario C can only be excluded if we know or assume that Monty always offers a choice. If we can't assume that Monty will always do that, then we can't exclude the possibility that scenario C may be invoked. In that case, the question is unanswerable, because you don't know if you'll improve your chances (scenario A), leave them unchanged (scenario B) or destroy them completely (scenario C).

If, however, we can assume, deduce or stipulate that Monty will always offer a choice, then scenario C can be excluded. In that case, even if we don't know whether scenario A or scenario B is in operation, then the answer is clear. Since switching cannot reduce your chances of winning, and may improve them (if A is the game in town), then switch anyway.

Rolfe.

 

[GreyICE]

C. "Monty-is-a-bastard" scenario, where he knows you have already chosen the correct door and is only offering you the switch to entice you away from the winning choice. Switching would lose you the car 100% of the time. However, as this is not a strategy (if employed consistently) that would produce a viable game show, it is arguable that it may be discounted.

And yet it is. The show is called "Lets make a deal" and Monty Hall is the host.

http://query.nytimes.com/gst/fullpage.html?res=9D0CEFDD1E3FF932A15754C0A967958260

After the 20 trials at the dining room table, the problem also captured Mr. Hall's imagination. He picked up a copy of Ms. vos Savant's original column, read it carefully, saw a loophole and then suggested more trials.

On the first, the contestant picked Door 1.

"That's too bad," Mr. Hall said, opening Door 1. "You've won a goat."

"But you didn't open another door yet or give me a chance to switch."

"Where does it say I have to let you switch every time? I'm the master of the show. Here, try it again."

On the second trial, the contestant again picked Door 1. Mr. Hall opened Door 3, revealing a goat. The contestant was about to switch to Door 2 when Mr. Hall pulled out a roll of bills.

"You're sure you want Door No. 2?" he asked. "Before I show you what's behind that door, I will give you $3,000 in cash not to switch to it."

"I'll switch to it."

"Three thousand dollars," Mr. Hall repeated, shifting into his famous cadence. "Cash. Cash money. It could be a car, but it could be a goat. Four thousand."

"I'll try the door."

"Forty-five hundred. Forty-seven. Forty-eight. My last offer: Five thousand dollars."

"Let's open the door."

"You just ended up with a goat," he said, opening the door. The Problem With the Problem

Mr. Hall continued: "Now do you see what happened there? The higher I got, the more you thought the car was behind Door 2. I wanted to con you into switching there, because I knew the car was behind 1. That's the kind of thing I can do when I'm in control of the game. You may think you have probability going for you when you follow the answer in her column, but there's the pyschological factor to consider."

He proceeded to prove his case by winning the next eight rounds. Whenever the contestant began with the wrong door, Mr. Hall promptly opened it and awarded the goat; whenever the contestant started out with the right door, Mr. Hall allowed him to switch doors and get another goat. The only way to win a car would have been to disregard Ms. vos Savant's advice and stick with the original door.

Monty Hall, it turns out, is also quite smart.

 

[Michael C]

And yet it is. The show is called "Lets make a deal" and Monty Hall is the host.

http://query.nytimes.com/gst/fullpage.html?res=9D0CEFDD1E3FF932A15754C0A967958260

Monty Hall, it turns out, is also quite smart.

Indeed, indeed. From that story, and from experience of other similar game shows, I conclude that Monty always knew where the car was and would not have used the random strategy: he liked to be able to control the outcome as much as possible.

In fact Rolfe missed one strategy in the previous post: the "Monty-is-super-nice" strategy: Monty only offers the chance to switch if there is a goat behind your chosen door. So once Monty has opened a door that reveals a goat, he can choose between the following strategies:

- "Classic" Monty: he always gives the chance to switch.

- "Super-nice" Monty: he only gives the chance to switch if there is a goat behind your door.

- "Bastard" Monty: he only gives the chance to switch if the car is behind your door.

If you play the strategy "always switch if the possibility is given", you will win 2/3 of all games against "classic" Monty, you will always win against "super-nice" Monty and you will never win against "bastard" Monty.

Since Monty will in fact mix the three strategies, there is no consistent way for you to increase your odds of winning.

 

[Ivor the Engineer]

<snip>

Since Monty will in fact mix the three strategies, there is no consistent way for you to increase your odds of winning.

That is unless you can estimate the probability of Monty using a particular strategy. For example, if it is known he selects between Classic (2/3), Nice (1) and Bastard (0) with equal probability, switching to the other door is still optimal (5/9).

 

[ddt]

Indeed, indeed. From that story, and from experience of other similar game shows, I conclude that Monty always knew where the car was and would not have used the random strategy: he liked to be able to control the outcome as much as possible.

IRL, I doubt that the presenter would not know where the car is. Even in case he'd forget, the director of the show behind the scenes would whisper it into his ear (piece).

Since Monty will in fact mix the three strategies, there is no consistent way for you to increase your odds of winning.

Not to mention that fact that Monty will try to influence the people into switching or not, after he has shown a second door. It's impossible to quantify that.

That is unless you can estimate the probability of Monty using a particular strategy. For example, if it is known he selects between Classic (2/3), Nice (1) and Bastard (0) with equal probability, switching to the other door is still optimal (5/9).

Which is still quite a way up from the 1/3 if you didn't have the whole switching business at all. Unfortunately, we don't get to see normally post-facto which door it had been (plus I don't care for this kind of shows).

BTW, Happy Birthday, Ivor!

 

[CurtC]

That is unless you can estimate the probability of Monty using a particular strategy. For example, if it is known he selects between Classic (2/3), Nice (1) and Bastard (0) with equal probability, switching to the other door is still optimal (5/9).

Seems to me that if he sometimes uses Nice and Bastard, then that rules out the Classic Monty, which requires that he always offer the switch. In other words, on those times when he's the Classic Monty, he's really either Nice or a Bastard depending on the contestant's initial choice.

If he selected between Nice and Bastard with equal probability, then the contestant who finds himself with the switch/stick decision still has a 2/3 chance of winning if he switches.

I think Monty's actual strategy was to keep 'em guessing, which would mean that he should offer the switch more often when the first guess was the car door, less often when it was the goat door. If Monty offers the switch twice as often to those who initially pick the car than he does to those who initially pick the goat (for example he will offer the switch 80% of the time to those who chose the car door and 40% of the time to those who chose the goat), then the poor sap contestant who is faced with the decision has a gut-wrenching 50/50 choice. But at least that's better than the 1/3 chance he had if Monty simply revealed the choice every time!

 

[Rolfe]

And yet it is. The show is called "Lets make a deal" and Monty Hall is the host.

http://query.nytimes.com/gst/fullpage.html?res=9D0CEFDD1E3FF932A15754C0A967958260

Monty Hall, it turns out, is also quite smart.

OK, fair enough, my premise was that this scenario would only be viable as part of a mixed run of scenario-switching, otherwise it would be far too predictable. But if you wanted to smack down a smart-aleck who hadn't taken all the possibilities into account and didn't think Monty would do that, then yes, it makes perfect sense.

Rolfe.

 

[Rolfe]

Indeed, indeed. From that story, and from experience of other similar game shows, I conclude that Monty always knew where the car was and would not have used the random strategy: he liked to be able to control the outcome as much as possible.

In fact Rolfe missed one strategy in the previous post: the "Monty-is-super-nice" strategy: Monty only offers the chance to switch if there is a goat behind your chosen door. So once Monty has opened a door that reveals a goat, he can choose between the following strategies:

- "Classic" Monty: he always gives the chance to switch.

- "Super-nice" Monty: he only gives the chance to switch if there is a goat behind your door.

- "Bastard" Monty: he only gives the chance to switch if the car is behind your door.

If you play the strategy "always switch if the possibility is given", you will win 2/3 of all games against "classic" Monty, you will always win against "super-nice" Monty and you will never win against "bastard" Monty.

Since Monty will in fact mix the three strategies, there is no consistent way for you to increase your odds of winning.

Yes, I think that in the real show (as opposed to the brain-teaser), it's unlikely that Monty would either not have known where the car was, or would have completely discounted that knowledge when deciding what to do next. It's mainly that that possibility is inherent in the brain-teaser as it is usually presented.

I was considering "super-nice" Monty as a subset of the classic scenario, but you're right, it's technically a different setup. However, given that Monty in the real game was never consistent, you'd never know if he was doing that, so you'd probably never decide to bank on it.

That is unless you can estimate the probability of Monty using a particular strategy. For example, if it is known he selects between Classic (2/3), Nice (1) and Bastard (0) with equal probability, switching to the other door is still optimal (5/9).

But once you're talking about real life rather then the brain-teaser, you can't even bank on Monty's distribution of strategies remaining constant. As in the example GreyICE quoted, if he thought someone had him sussed, he'd just swing to a different approach to confound them. Sort of like Curt C. was suggesting. Sweet.

Rolfe.

 

[ZirconBlue]

Thank you.
I am forever in debt to you.
The whole derail was because of that sentence.

*kisses zirconblue all over*

Don't mention it.