Monty Hall Problem

[sol invictus]

Yes but he must show a door and the door will be a goat... and so your odds are always doubled from 1/3 (by staying) to 2/3 by switching. You don't have to know what door to switch to... just that you will switch to the door he doesn't open--because you can't choose the one he does (nor would you want to... since in the situation stipulated, it's always a goat.)

No offense, but I'm going to bow out of this exchange now. As far as I can tell there is zero information going in either direction.

 

[articulett]

That's fine. I'm done too. I provided links... you can see the classic odds in the first link...

(you can play as many versions of the game as you want) Or you can try the alternating scenarios (blind host, etc. in the second link.

I realize that Claus will never explain what he means by saying your odds change when the host reveals a goat. The only thing that changes.... is that you can't or won't pick that door (the one the host revealed)... that means the remaining door has 2/3 of a chance of having the car behind it. Your original choice still has 1/3

 

[CFLarsen]

That's fine. I'm done too. I provided links... you can see the classic odds in the first link...

(you can play as many versions of the game as you want) Or you can try the alternating scenarios (blind host, etc. in the second link.

I realize that Claus will never explain what he means by saying your odds change when the host reveals a goat. The only thing that changes.... is that you can't or won't pick that door (the one the host revealed)... that means the remaining door has 2/3 of a chance of having the car behind it. Your original choice still has 1/3

Where did I say that the odds suddenly became 50-50? Are you ever going to provide evidence of that claim?

 

[RecoveringYuppy]

Try the links... you'll see...

FWIW I'm sure he sees. I'm fairly sure Claus gets it too, but I haven't read every detail in his posts. It appears to me you (articulett) are being obtuse in your wording in order to inject "Claus is wrong" phrases.

It's obtuse to object to someone saying that conditions change, or that probabilities changes as conditions (known information) changes.

It's also obtuse to object to someone pointing out what is and isn't the Monty Hall problem. As with this thread, confusion over a nonintuitive idea isn't helped by a misstatement of the problem.

Take a break.

 

[GreyICE]

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

Claus, you are one of the least likable human beings its ever been my distinct displeasure to have discourse with on the internets. I don't know what it is. We have many opinions in common, you seem reasonably well educated, you don't troll, flame, or otherwise behave blatantly poorly, and therefore I am at a loss to explain why, three posts into any thread you're posting in, I want to strangle you with a garden hose. 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, are one of the most passive-aggressive people I've ever interacted with, have the logical capacity of a rhubarb, and generally have all the sense of proportion of your average 16 year old.

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?

 

[Michael C]

Things to get straight:

1. The odds of having picked the correct door the first time never change.

2. The odds of winning can change at some point in the game.

In the case of the "classic" Monty Hall problem, the odds of winning do not change at the point where Monty opens the door with a goat: they are still 1/3. They can change at the point where Monty offers the choice of switching doors, if the contestant chooses to switch.

In the case of the variant of the problem where Monty chooses a door at random and there happens to be a goat behind it, the odds of winning do change when Monty opens this door: they change from 1/3 to 1/2.

 

[sol invictus]

Things to get straight:

1. The odds of having picked the correct door the first time never change.
<snip>
In the case of the variant of the problem where Monty chooses a door at random and there happens to be a goat behind it, the odds of winning do change when Monty opens this door: they change from 1/3 to 1/2.

These two statements appear to contradict each other. At best, they are sufficiently imprecise that one can't tell for sure.

Statement 1 is correct only if:

a) it's intended to apply only to the standard variant, or

b) you mean the odds with no conditions or extra information from opening doors applied - in which case it's a tautology, since the odds trivially cannot change.

It's extremely confusing to say "the odds never change" when you're talking about something which by definition cannot change. It's like saying "the weather at noon on April 12 1963 never changes".

 

[Michael C]

These two statements appear to contradict each other. At best, they are sufficiently imprecise that one can't tell for sure.

Statement 1 is correct only if:

a) it's intended to apply only to the standard variant, or

b) you mean the odds with no conditions or extra information from opening doors applied - in which case it's a tautology, since the odds trivially cannot change.

It's extremely confusing to say "the odds never change" when you're talking about something which by definition cannot change. It's like saying "the weather at noon on April 12 1963 never changes".

I mean "b": the opening odds. Yes, I agree that it's confusing to say "the odds never change" if we are talking about something which by definition cannot change. So in statement 1, I agree that I am stating the obvious.

This, however, seems to be the main point of Articulett. It confused me until I realised that she really was talking about the beginning odds of 1/3, which by definition never change, and I was talking about the odds of winning calculated at a given point in time, which change throughout the game. The odds of winning start at 1/3, and at the very end they are either 1 or 0; in between they may change to 2/3, or 1/2, or something else, depending on what version of the game Monty is playing and what choices you make.

 

[Herzblut]

These two statements appear to contradict each other. At best, they are sufficiently imprecise that one can't tell for sure.

Right. Articulett doesn't know statistics or statistical lingo, hence she's talking in mysterious terms like "the odds I have chosen noodles for dinner yesterday". This might help her to understand a problem, but it doesn't allow for a meaningful discussion.

What she arguably means, in proper terms, is the marginal probability or prior probability. Which is, of course, not what the problem is all about. It's about optimizing winning chances in a concrete game situation. This is what playing a game is about anyway.

Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written P(A|B), and is read "the probability of A, given B".

Marginal probability is then the unconditional probability P(A) of the event A; that is, the probability of A, regardless of whether an event B did or did not occur.

Joint probability the probability of two events in conjunction. That is, it is the probability of both events together. The joint probability of A and B is written

or

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

A prior probability is a marginal probability, interpreted as a description of what is known about a variable in the absence of some evidence. The posterior probability is then the conditional probability of the variable taking the evidence into account.

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

The problem in question, of course, tasks us with determining a posterior/conditional probability, the probability of winning by switching or not switching, given Monty has opened a goat door. Obviously, the prior probability of winning by not switching is 1/3. I've shown how to calculate the posterior probability by simple, clear mathematics. I encourage the reader to have the heart to take a look at the mathematical concept, which is very simple and elegant, two lines of calculation replace 1000 lines of needless discussion.

For further discussion, if any, we might use the correct terms prior probability (before opening a door) and posterior probability (after a door has been opened), because they are clearly defined and nicely descriptive.

 

[Rolfe]

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

Claus, you are one of the least likable human beings its ever been my distinct displeasure to have discourse with on the internets. I don't know what it is. We have many opinions in common, you seem reasonably well educated, you don't troll, flame, or otherwise behave blatantly poorly, and therefore I am at a loss to explain why, three posts into any thread you're posting in, I want to strangle you with a garden hose. 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, are one of the most passive-aggressive people I've ever interacted with, have the logical capacity of a rhubarb, and generally have all the sense of proportion of your average 16 year old.

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?

Rolfe nods enthusiastically, entirely in agreement with GreyICE on every point.

 

[Rolfe]

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

According to Wikipedia, the first publication of the puzzle dates back to 1975. (When did the Monty Hall TV show start, does anyone know?) 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.

However, as far as I can see, the perennial fascination owes a great deal to the fact that subsequent wordings of the scenario have become more ambiguous. I can't see how a puzzle clearly and precisely defined as above could be anything more than a nine-day wonder.

Certainly, when I first encountered the puzzle in 1994 or thereabouts, the wording was similar to the OP, and arguably ambiguous. And then, as now, the bulk of the argumentation seemed to revolve around exactly what rule Monty is to be supposed to be working to. And to be honest, I think this raises the debate to a whole new level.

If we just consider what Claus and CurtC and others call the "classic Monty Hall" scenario, yes, it's intriguing, and counterintuitive, but the answer is perfectly clear once you've got your brain around it. I think it took me a day or two, just puzzling at it on my own, to get there. Really, really, even though there might be some dimwits who will never get it, there isn't a debate. Switching doubles your chances of getting the car, full stop. End of.

All everybody has been arguing about for most of this thread is interpretation. Exactly what game are we playing anyway?

I know how I got on to that aspect, and I think it's how many people have got there. I imagined the 100-doors version (99 goats and a car), while I was thinking about the basic puzzle. This illustrates how the odds change during the process, but it also clarifies the need to know what the exact rules are.

Contestant chooses a door. Monty opens another, a goat. Switch? Wouldn't make much difference, you've still got 98 other doors. But as the process is repeated and repeated, the situation becomes much clearer. The longer the car remains unrevealed, the clearer it becomes that Monty is deliberately avoiding it. By the time there are only two doors left, two things are obvious. The probability that Monty is deliberately avoiding the car is 98/100. And it should be completely intuitive that you should switch. Because the probability that the car is behind the door you originally chose is and always was 1/100, therefore the probability that it is behind the other door is 99/100.

For me, it was this exercise that clarified the need to stipulate the exact rules. Because in the 100-doors version, if Monty was opening doors at random, with no more idea than you have of where the car is, then in the overwhelming majority of games, he'd have revealed the car behind one of the other doors way before you're left with only two closed doors.

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.

Contestant picks a door. Monty, who has no idea where the car is, opens one of the other two doors at random. A third of the time he reveals the car. Oh too bad sir, thanks for playing, good game, hope you and your goat will be very happy together. Two thirds of the time he reveals a goat. Well sir, so far so good, but would you like to change your choice? In that situation, switching makes bugger-all difference.

Cue argument that lasts for (to date) 33 years.

But then it gets worse. Now that the matter of the exact rules has been opened to scrutiny, people start getting even more creative. What if Monty is a complete bastard, and will only open the door if he knows you've already got the car? Well, unlike the random-opening scenario, that is neither a sensible game show, or a reasonable brain-teaser. As one of a range of possibilities incorporated in my next scenario (capricious Monty) it has a place. However, as a consistent stratagem, it's a nonsense. It's just a hypothetical scenario that is dragged in for the sake of argument.

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.

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.

For these reasons I maintain that it is reasonable to exclude both the "Monty-is-a-bastard" scenario and the "Monty-is-capricious" scenario. No game show could possibly persist if the former scenario was being operated, and no rational brain-teaser could employ the latter.

Nevertheless, what I will call the "B" scenario, the one where Monty doesn't know where the car is but simply opens one of the two doors the contestant hasn't chosen, is valid as a brain teaser and as a game show, and is in accordance with the usual, ambiguous, wording of the puzzle. (I'm designating the "real" Monty Hall puzzle, where he knows where the car is and will avoid opening that door at this stage, as the "A" scenario.)

This is the conundrum I find completely and utterly fascinating.

The puzzle is as presented. It is not clear whether the A or the B scenario is intended. (That is, you deduce that Monty opening one of the doors you didn't choose is a prerequisite, but you don't know whether or not he knows where the car is.) Where does that leave you?

It leaves you in the peculiar position where the correct answer depends entirely on what is going on in the mind of the host. And in my experience it is that very surreal situation that causes the most outcry from those who have difficulty with the puzzle.

I can only really get my brain round it by imagining that I'm playing one of two computer-game versions. One is programmed with the "A" version, and the other with the "B". I've got to the point where one of the doors I didn't choose has been opened, and there is a goat there, but I still don't know which one I'm playing. If it's the "A" version, I know that switching will double my chances of winning, but if it's the "B" version, I know that switching won't make a blind bit of difference.

Seems sensible enough when looked at that way.

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.

 

[articulett]

I want to point out something that might clarify it for those who haven't got it.

If the host is "blind" (doesn't know what he's choosing)-- then you are not doing the Classic Monty Hall problem (which stipulates that a goat is always shown.)

The same is true in the case where Monty only offers the choice when you have the car (because the classic situation says he always offer the choice.)

However, whether you are involved in the classic Monty Hall situation or have no knowledge as to which of the 3 options you are dealing with-- on average, your odds double from 1/3 to 2/3 if you are offered to switch and do so. If the host is in one of the latter 2 categories-- you will either have 1/3 of a chance of having your choice taken away because the host accidentally reveals a car (blind Monty) OR you will have a 2/3 chance of having your choice taking away, because the choice is only offered if you have car hidden behind the door you chose.

The odds aren't changing... just your odds of being offered a choice.

 

[articulett]

Where did I say that the odds suddenly became 50-50? Are you ever going to provide evidence of that claim?

I'll show when you explain exactly what you meant when you insisted that "the odds change when the host offers you the option to switch." What odds change to what, Claus? You were adamant about this... and said I was wrong to say they don't.

 

[articulett]

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

Claus, you are one of the least likable human beings its ever been my distinct displeasure to have discourse with on the internets. I don't know what it is. We have many opinions in common, you seem reasonably well educated, you don't troll, flame, or otherwise behave blatantly poorly, and therefore I am at a loss to explain why, three posts into any thread you're posting in, I want to strangle you with a garden hose. 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, are one of the most passive-aggressive people I've ever interacted with, have the logical capacity of a rhubarb, and generally have all the sense of proportion of your average 16 year old.

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?

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.

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.

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.

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.

 

[articulett]

FWIW I'm sure he sees. I'm fairly sure Claus gets it too, but I haven't read every detail in his posts. It appears to me you (articulett) are being obtuse in your wording in order to inject "Claus is wrong" phrases.

It's obtuse to object to someone saying that conditions change, or that probabilities changes as conditions (known information) changes.

It's also obtuse to object to someone pointing out what is and isn't the Monty Hall problem. As with this thread, confusion over a nonintuitive idea isn't helped by a misstatement of the problem.

Take a break.

He was the one who claimed I was wrong and didn't know anything about the problem. What is "obtuse" about my wording? Highlight it and tell me how you would have explained it differently. And what odds change to what --as per Claus' claim that "your odds change when the host offers you an option to choose"? No odds NOR conditions change-- it is stipulated that you are always shown a goat and always offered a switch. He made this claim to assert I was wrong. So what are these odds that change exactly?

 

[TrueSceptic]

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

According to Wikipedia, the first publication of the puzzle dates back to 1975. (When did the Monty Hall TV show start, does anyone know?) 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.

However, as far as I can see, the perennial fascination owes a great deal to the fact that subsequent wordings of the scenario have become more ambiguous. I can't see how a puzzle clearly and precisely defined as above could be anything more than a nine-day wonder.

Certainly, when I first encountered the puzzle in 1994 or thereabouts, the wording was similar to the OP, and arguably ambiguous. And then, as now, the bulk of the argumentation seemed to revolve around exactly what rule Monty is to be supposed to be working to. And to be honest, I think this raises the debate to a whole new level.

If we just consider what Claus and CurtC and others call the "classic Monty Hall" scenario, yes, it's intriguing, and counterintuitive, but the answer is perfectly clear once you've got your brain around it. I think it took me a day or two, just puzzling at it on my own, to get there. Really, really, even though there might be some dimwits who will never get it, there isn't a debate. Switching doubles your chances of getting the car, full stop. End of.

All everybody has been arguing about for most of this thread is interpretation. Exactly what game are we playing anyway?

I know how I got on to that aspect, and I think it's how many people have got there. I imagined the 100-doors version (99 goats and a car), while I was thinking about the basic puzzle. This illustrates how the odds change during the process, but it also clarifies the need to know what the exact rules are.

Contestant chooses a door. Monty opens another, a goat. Switch? Wouldn't make much difference, you've still got 98 other doors. But as the process is repeated and repeated, the situation becomes much clearer. The longer the car remains unrevealed, the clearer it becomes that Monty is deliberately avoiding it. By the time there are only two doors left, two things are obvious. The probability that Monty is deliberately avoiding the car is 98/100. And it should be completely intuitive that you should switch. Because the probability that the car is behind the door you originally chose is and always was 1/100, therefore the probability that it is behind the other door is 99/100.

For me, it was this exercise that clarified the need to stipulate the exact rules. Because in the 100-doors version, if Monty was opening doors at random, with no more idea than you have of where the car is, then in the overwhelming majority of games, he'd have revealed the car behind one of the other doors way before you're left with only two closed doors.

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.

Contestant picks a door. Monty, who has no idea where the car is, opens one of the other two doors at random. A third of the time he reveals the car. Oh too bad sir, thanks for playing, good game, hope you and your goat will be very happy together. Two thirds of the time he reveals a goat. Well sir, so far so good, but would you like to change your choice? In that situation, switching makes bugger-all difference.

Cue argument that lasts for (to date) 33 years.

But then it gets worse. Now that the matter of the exact rules has been opened to scrutiny, people start getting even more creative. What if Monty is a complete bastard, and will only open the door if he knows you've already got the car? Well, unlike the random-opening scenario, that is neither a sensible game show, or a reasonable brain-teaser. As one of a range of possibilities incorporated in my next scenario (capricious Monty) it has a place. However, as a consistent stratagem, it's a nonsense. It's just a hypothetical scenario that is dragged in for the sake of argument.

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.

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.

For these reasons I maintain that it is reasonable to exclude both the "Monty-is-a-bastard" scenario and the "Monty-is-capricious" scenario. No game show could possibly persist if the former scenario was being operated, and no rational brain-teaser could employ the latter.

Nevertheless, what I will call the "B" scenario, the one where Monty doesn't know where the car is but simply opens one of the two doors the contestant hasn't chosen, is valid as a brain teaser and as a game show, and is in accordance with the usual, ambiguous, wording of the puzzle. (I'm designating the "real" Monty Hall puzzle, where he knows where the car is and will avoid opening that door at this stage, as the "A" scenario.)

This is the conundrum I find completely and utterly fascinating.

The puzzle is as presented. It is not clear whether the A or the B scenario is intended. (That is, you deduce that Monty opening one of the doors you didn't choose is a prerequisite, but you don't know whether or not he knows where the car is.) Where does that leave you?

It leaves you in the peculiar position where the correct answer depends entirely on what is going on in the mind of the host. And in my experience it is that very surreal situation that causes the most outcry from those who have difficulty with the puzzle.

I can only really get my brain round it by imagining that I'm playing one of two computer-game versions. One is programmed with the "A" version, and the other with the "B". I've got to the point where one of the doors I didn't choose has been opened, and there is a goat there, but I still don't know which one I'm playing. If it's the "A" version, I know that switching will double my chances of winning, but if it's the "B" version, I know that switching won't make a blind bit of difference.

Seems sensible enough when looked at that way.

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.

Just saw this thread, which has a life beyond all reason or probability.

Just to follow on from your post, I heard about this only a few years ago. The rules were not explained, only what happened in the show, so the accepted answer of switching increasing the probability from 1/3 to 2/3 was not at all obvious. To make this the case, one has to assume that Monty not only knows where the car is but is also obliged to open one door. As you say, the puzzle changes if either of those do not apply.

To take this further, if we assume that either your A or B apply but we don't know which, surely we should still switch because not switching leaves us with a 1/3 or 1/2 chance, whereas switching gives us 1/2 or 2/3, i.e., improving our odds?

 

[Rolfe]

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, U 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.

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.

 

[Rolfe]

To take this further, if we assume that either your A or B apply but we don't know which, surely we should still switch because not switching leaves us with a 1/3 or 1/2 chance, whereas switching gives us 1/2 or 2/3, i.e., improving our odds?

Yes, absolutely!

This now gets me back to my original point, which is that excluding a rigged game, you should switch. Because even if you're on the latter programme, switching won't decrease your chances. But there is a less-than zero possibility that you are on the former programme, when switching is beneficial. Therefore you should rationally decide to switch, to allow for the possibility that the former is the game in town.

I thiink this is what I like about it. The surreal conclusion that Monty's very intent affects the odds (made a bit less surreal if we substitute Monty with a pre-programmed computer game). And then, the extra leap that says, stop arguing. If you don't know which scenario you're dealing with, and in one the switch is beneficial while in the other it's neutral, then just switch anyway - you can't lose by it, and by switching you are in a position to exploit the possibility of the beneficial scenario.

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".

I said this about six times already, but as far as I can see, you're the only person who has read any of my posts recently.

Rolfe.

 

[articulett]

Just saw this thread, which has a life beyond all reason or probability.

Just to follow on from your post, I heard about this only a few years ago. The rules were not explained, only what happened in the show, so the accepted answer of switching increasing the probability from 1/3 to 2/3 was not at all obvious. To make this the case, one has to assume that Monty not only knows where the car is but is also obliged to open one door. As you say, the puzzle changes if either of those do not apply.

To take this further, if we assume that either your A or B apply but we don't know which, surely we should still switch because not switching leaves us with a 1/3 or 1/2 chance, whereas switching gives us 1/2 or 2/3, i.e., improving our odds?

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.

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-- .)

 

[TrueSceptic]

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".

I said this about six times already, but as far as I can see, you're the only person who has read any of my posts recently.

Rolfe.

As I said, this thread has only just got my attention. I read only the first few and last few posts and the post I replied to seemed to include all the options that whizz around my brain whenever I think about the problem. I confess to focussing on the tail of your post because it is the nub of the puzzle IMO. Anything else is not the Monty Hall Problem as I see it, and Articulett seems of the same view.