Talk:Monty Hall problem/Archive 19

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Falk

In the 'Sources of confusion' section we currently have the following:

A competing deeply rooted intuition at work in the Monty Hall problem is the belief that exposing information that is already known does not affect probabilities (Falk 1992:207). This intuition is the basis of solutions to the problem that assert the host's action of opening a door does not change the player's initial 1/3 chance of selecting the car. For the fully explicit problem this intuition leads to the correct numerical answer, 2/3 chance of winning the car by switching, but leads to the same solution for slightly modified problems where this answer is not correct (Falk 1992:207).

It is not clear to me what this is intended to mean. Is Falk saying, and are we agreeing, that exposing information that is already known does affect probabilities? Martin Hogbin (talk) 22:11, 29 June 2010 (UTC)

We talked about this before. She's saying that solutions that say things like "since we already know one of the two unopened doors must be a goat, opening one and showing a goat cannot affect the probability of the originally picked door being the car" appeal to this notion, but this reasoning is no more valid than the naive "there are two doors left so they must have equal probabilities" reasoning. Both of these are based on deeply held intuitions. Neither are valid reasoning in this case. Rick Block (talk) 22:25, 29 June 2010 (UTC)

But opening a specific door can reveal information if the host has a known door preference, as we all know, thus the comment attributed here to Falk would appear to be something of a non-sequitur.

Do we all agree that if no information is revealed, the probability stays the same, but if information is revealed, the probability can change. Martin Hogbin (talk) 22:50, 29 June 2010 (UTC)

Where we talked about this before was /Archive 16#Information?. I suggested deleting "the belief". Would that address whatever your concern is? Rick Block (talk) 00:53, 30 June 2010 (UTC)

My main concern is that I cannot understand what point we are trying to make, deleting "the belief" will not make it any clearer to me. What is you answer to my question above? Martin Hogbin (talk) 01:30, 30 June 2010 (UTC)

@Martin: Agree. But: probability to whom?? To the guest, to any chiefly kibitzer ignoring and destorting the rules, or to the host, or to whom? Important to read the rules:

The host never opens "two" doors and he never opens even "1,01" doors, but he strictly is opening only one single door. Only one. Thats the essence of the rules.

I do not understand the point that you are making here. Who claims the host ever opens more than one door? Martin Hogbin (talk) 12:14, 30 June 2010 (UTC)

As I (yesterday 18:05 above) wrote to Rick:

The cars problem is different? source? Switching wins in 2/3 of all cases. punctum. Sure, the host has all information on hand, but he never can change the actual constellation. All he can (eventually??) do, is giving additional info on the actual constellation: e.g., if opening his "unwanted" door (q=0 or 1) in 1/3 of all cases, showing then that his preferred door DOES hide the car (as if he was opening it, too) and, when opening his preferred door, the chances of both still closed doors are 1/2 each. But this is not part of the dilemma. This is a totally different issue. Yes, knowing the Host's bias would be great, but the guest does NOT know it and she should switch anyway! Only a "known and quantified door preference q (off-size 1/2)" could generate a possible "condition" in opening any door, giving a closer hint on the actual constellation. In the absence of such exact knowledge any semi-hidden assumption is unhelpful, confusing, distracting and real bad obfuscation. So-called "Conditional" in the absence of any evident condition cannot give any closer "conditional hint" on the respective actual and given situation. To the guest, the absence of a known and evident condition makes "conditional" solutions cryptical and worthless. No more semi-hidden and confusing assumptions, please. Claiming "conditional" without any evident condition. Such puzzlements are not part of the famous dilemma. Or name them frankly what they are and put them in a separate section, at least. Such meander do not address the famous dilemma. They are a quite different issue. (end)

If the host ("q=0 or q=1") opens the strictly avoided door showing a goat, then he evidently is demonstrating (showing) that his preferred door actually hides the car (P=1), like opening two or three doors at the same time, thus the door selected by the player definitely is a goat and NOT THE CAR (P=0). This is not provided for in the original MHP-question. Any variant letting the host show the contents of say 1 1/40, 1 1/4, 1 1/2 or even two or three doors is a far distant farcical variant, never addressing the original MHP-question, even if that semi-hidden discrepancy remains disguised by confusing weasel words. --Gerhardvalentin (talk) 14:19, 30 June 2010 (UTC)

Once more: in the MHP the host opens only one door, never two doors (that would be the case in 1/3 of all cases with "alleged q=0 or q=1"). That means in English: No so-called alleged "door-preference" whatsoever of the host. If no reliable quantification of a "supposed" host's door-preference "q" is evident and exactly known (then it was not the "MHP" any more, see the rules), i.e. even if "q is unknown" or, like the rules say: "q=1/2") then by opening of a specific door no additional information whatsoever regarding the chance of the originally selected door is revealed. Thus: No new condition. Read the rules. So, by opening of only one door, conditions remain totally unchanged. Otherwise it wasn't the "MHP". Even if "reliable sources" offer semi-hidden assumptions, weaslewording and destorting the rules, amidst applause from the crowd. --Gerhardvalentin (talk) 01:53, 30 June 2010 (UTC)

Are you pointing out here that any host preference is unknown to the player? Martin Hogbin (talk) 12:14, 30 June 2010 (UTC)

Yes. The original MHP-question never provides for any host's bias. The host only shows the content of one single door, nothing else. So the player cannot have knowledge of any such alleged bias. And the player is the instance to judge about odds and probabilities, the question is asked based on the player's knowledge, not to s.o. who in private makes the rules himself. Any "desired" host's bias is not part of the MHP, but is welcome to students of probability theory for training purpose. But never addressing the MHP-question. Even though this sometimes has inaccurately been claimed by some.--Gerhardvalentin (talk) 14:19, 30 June 2010 (UTC)

It looks like we agree completely then. For some reason people are happy to take the car's initial placement as uniform, despite this not being stated in the question, but will not take the host's goat door choice to be uniform, even though the question tells us nothing about that either. This is not the point that I am talking about here though. Martin Hogbin (talk) 15:53, 30 June 2010 (UTC)

@Martin - I don't understand what you don't understand. The point is that solutions that say things like "since we already know one of the two unopened doors must be a goat, opening one and showing a goat cannot affect the probability of the originally picked door being the car" are intuitively appealing, but are mathematically, um, what's the right word, false. It IS true that exposing information that is already known does not affect probabilities - but what is not true is that opening a specific door showing a goat knowing at least one of two doors hides a goat exposes only information that is already known. Consider the "Monty crawls" variant. Or the "Monty forgets" variant. You already knew there was a goat behind one of the other two doors in these variants as well. In the standard version it is true no information is revealed. But this is something to be concluded based on some reasoning - and the reason is clearly not simply that you already knew there was a goat behind one of two doors. Perhaps you or Gill would like to discuss this with Gerhardvalentin or Glkanter. -- Rick Block (talk) 03:40, 30 June 2010 (UTC)

Let me start with some things that we do agree then:

1. If no new information is revealed when a door is opened then probability of the originally picked door being the car does not change.
2. In the standard version of the problem, no information is revealed when a door is opened.

Are we agreed so far?

Your point is that 2 should not be regarded as obvious, since there can be apparently similar circumstances when it is not true. Is that right? Martin Hogbin (talk) 12:06, 30 June 2010 (UTC)

This depends on what "information" means. New information is revealed in the sense that first you did not know which door the host is going to open, and then you do. New information "about door 1" is not revealed, in the sense that the probability that the car is behind that door is not changed by the revelation (you can also phrase this in terms of statistical independence, because P(A | B) = P(A) is equivalent to P(A, B) = P(A)P(B), simply multiply by P(B) on both sides; "no information about" is usually formalized as a (conditional) independence assumption, and AFAICS this is the sense of "information" you are thinking of). See this page from Rosenhouse, which uses the first sense of information: "information is received that decreases the probability of one of those events to 0", "We have received new information that may, or may not, give us reason to alter our assessments of the probabilities." Rosenhouse also criticizes assuming that P(A | B) = P(A) by saying that "Sadly, this merely assumes what we are trying to prove. [...] So we cannot merely assert that Monty's actions fail to alter the probabilities. That must be proved." -- Coffee2theorems (talk) 14:37, 30 June 2010 (UTC)

I think we all agree then. Let me repeat what I said more precisely put:

1. If, when a door is opened, no new information is revealed about the probability that the car is behind door 1, then the probability that the car is behind door 1 does not change when a door is opened .
2. In the standard version of the problem, no information about the probability that the car is behind door 1 is revealed when a door is opened.
3. Many people think that point 2 is not obvious.

Do we all agree with that? Martin Hogbin (talk) 14:52, 30 June 2010 (UTC)

Excellent. That's it, you brought it to the point: Your wording "point 3" flashes the reason of the popular 50:50 error, and it simultaneously names exactly the subject that bias-conditionalists are pondering about. --Gerhardvalentin (talk) 15:21, 30 June 2010 (UTC)

I agree with the intent of the above, but as a minor nit about wording, "information about the probability that the car is behind door 1" should be replaced with "information about the car being behind door 1". It's information about the random variable X = "car is behind door 1", not about a probability P(X | ...). Maybe this clarifies it: Let X and Y be two independent random variables (e.g. standard normal). Knowing Y does not give you any information about X. In the MHP, Y = "host opens door 3" and X = "car is behind door 1". -- Coffee2theorems (talk) 16:06, 30 June 2010 (UTC)

Thank you, your wording is better, but apart from that we all agree. The question is, what is Falk saying and what are we saying. The confusion we are trying to find a source for is that people think that, after a door has been opened, the probability that the car is behind the original door is 1/2 and the probability it is behind door 2 is also 1/2. How does our statement explain that? Martin Hogbin (talk) 17:23, 30 June 2010 (UTC)

Is he talking about a "biased host" (q=0 or q=1)? In 1/3 of all cases the host disposes over the car and one goat, but the goat being behind his preferred door. So he is forced to open his "avoided door", and by that showing that the car is behind his preferred door. Chance of the offered door=1, chance of the door selected by the guest=0. As said this is 1/3 of all cases.

In another 1/3 of all cases the host, disposing over the car and one goat, the car being behind his "avoided door", he can open his preferred door. And in 1/3 of all cases, having to chose between two goats, he also can open his preferred door. So in 2/3 of all cases the chance of both unopened doors are 1/2 each. --Gerhardvalentin (talk) 22:08, 30 June 2010 (UTC)

When I first heard of the MHP-question from a friend I also answered the chances of the two still closed doors are 1/2 to 1/2. - Why? Two closed doors, one of them must be the car, and the other one must be the second goat, so "50:50". May be the real problem is how the question is put (see this). If the guest had the chance to chose either one door or two doors, knowing that only one car is provided for, everything was clear. Or if the host would offer both unchosen but still unopened doors for a switching. But the crux is that the host first opens one door before he offers the chance to switch "to the two unselected doors". This gave room for improper and fruitless wisenheimer speculations about "conditional probability" without any relevant condition, amidst applause from the crowd (remember: "no biased host"!). -Gerhardvalentin (talk) 21:45, 30 June 2010 (UTC)

Some interesting solutions there.

Unless someone can explain what the statement attributed to Falk means and how it addresses the causes of confusion in the MHP, that is to say why people thing the answer is 1/2 rather than 2/3, I propose to delete it. Martin Hogbin (talk) 22:49, 30 June 2010 (UTC)

@Martin - You seem to be completely missing the point. The confusion here is not that the probability of winning by switching is 1/2, but that the probability is 2/3 (!) based on the reasoning that the host opening a door doesn't change the original 1/3 because we already know one of the two unchosen doors is a goat. Falk presents these views (two unknowns must have equal probability vs. the host opening a door doesn't change the original 1/3) as the two typical solutions - but is saying while both are intuitively appealing they are both based on erroneous reasoning. Although the second one ends up with the correct numeric answer for the standard version of the problem, the same reasoning fails for slight variants (like the Monty crawls, or Monty forgets variant) - i.e. this reasoning is not sound. -- Rick Block (talk) 04:57, 1 July 2010 (UTC)

Rick: You referred to the essential aspects in a clear and concise form. It was fine if the article could show those aspects in the same clear and coherent manner. --Gerhardvalentin (talk) 06:59, 1 July 2010 (UTC)

@Rick - Firstly, it is not clear to me at least (and I have been interested in this problem for some time) that this is what the paragraph is all about. It could do with some clarification.

Secondly, the 'confusion' to which the rest of the section refers is the fact that people thing the answer is 1/2. This is the confusion that caused vos Savant to receive thousands of letters and it is the confusion that makes the problem notable. What you are referring to is the reason that some sources insist that the problem must be treated conditionally. This belongs in a different place, if anywhere. As it is now it is itself a source of confusion. Martin Hogbin (talk) 09:09, 1 July 2010 (UTC)

How about having this somewhere in the 'Conditional probability' section:

A competing deeply rooted intuition at work in the Monty Hall problem is the belief that when the host opens a door he is exposing information that is already known and that this does not affect the probability that the car is behind the door originally chosen by the player. (Falk 1992:207). This intuition is the basis of solutions to the problem that assert the host's action of opening a door does not change the player's initial 1/3 chance of selecting the car. For the fully explicit problem this intuition leads to the correct numerical answer, 2/3 chance of winning the car by switching, but leads to the same solution for slightly modified problems where this answer is not correct (Falk 1992:207).

If this is presented in a distant section on the "Conditional solution" to what does "competing" refer? The essential point here is that solutions based on appeal to intuition rather than a careful probabilistic analysis are likely to be wrong. These two intuitions are both confusing and both lead to irrational attachments to incorrect solutions. -- Rick Block (talk) 04:01, 2 July 2010 (UTC)

Rick Block, you are, without doubt, the worst analyst, and worst paraphraser I have ever encountered. Your conclusions are oftentimes blindingly erroneous, pointless, and wholly unsupported by the 'sources'. This one takes the cake, though. Glkanter (talk) 06:28, 2 July 2010 (UTC)

Rick, perhaps you could post some direct quotes of the relevant sections of Falk on the /Sources page for discussion. I am sure that there will be no copyright problems in copying sections of a paper for private discussion. If we cannot agree on what point is being made and what in this article the paper relates to, the section should be removed. Martin Hogbin (talk) 08:44, 2 July 2010 (UTC)

Propose removal

There is no consensus as to why this paragraph is in the article is or what it even means. I propose to remove it unless somebody can give a clear indication of its exact meaning and purpose. Martin Hogbin (talk) 09:29, 4 July 2010 (UTC)

What's Your Beef, Glopk?

How is my edit to the Simple section not within Wikipedia's guidelines? Why did you revert it? Glkanter (talk) 16:51, 30 June 2010 (UTC)

I agree. When an editor's work is reverted, some kind of explanation is in order. Martin Hogbin (talk) 17:24, 30 June 2010 (UTC)

Explanation: see previous reverts of the same change. And please do leave the beef alone, am vegetarian. glopk (talk) 17:42, 30 June 2010 (UTC)

How's about we discuss it with full sentences and paragraphs? Besides, that was before, when Carton's solution was presented as a table, not a decision tree. What's your problem with it?

You mean more full sentences and paragraphs than there are in the section "Carlton decision tree" above? My problem is your repeated introduction of that picture despite the obvious lack of editorial consensus, and a good argument that it is OR . glopk (talk) 21:41, 30 June 2010 (UTC)

Yeah, I don't see your contribution to the discussion anywhere in that section. Or this one where Rick Block and Coffee2theorems show in post after excruciating post how little they understand such a simple table.

So, it's nice that you're the 'big brother' looking out for his weaker siblings.

Looking out for the FA-class article, actually. And no, I don't have brothers either big or small. Your imaginary world looks like an interesting place, do you sell tickets for casual visitors? glopk (talk) 16:30, 1 July 2010 (UTC)

I look out for the FA-class article as well. Here was my 1st edit to the MHP in October, 2008. I deleted this from the very first line of the Solution section:

"The overall probability of winning by switching is determined by the location of the car."

Really. That was after the 2nd FAR. Here's the diff. Glkanter (talk) 16:54, 1 July 2010 (UTC)

Of course, the 'consensus' is being denied by Morgan-disciple Rick Block, who has been wrong on just about everything MHP-related for the nearly 2 years I've been in this discussion, and Coffee2theorems who still doesn't understand the solution. I disagree with your argument that the decision tree is OR. Everything comes from Carlton of Whitaker via vos Savant. But, that's a matter of opinion/interpretation, I guess.

Since you're the one that did some reverts, I think it's important and reasonable to know what *you* think Glopk, of Carlton's decision tree? Does the math stand up? Does it show that the random host premise is unnecessary? Glkanter (talk) 23:02, 30 June 2010 (UTC)

Hey, Glopk, what about the text? Why is some muddled paraphrasing superior to the actual 3 sentence solution Carlton gives? Glkanter (talk) 23:26, 30 June 2010 (UTC)

Please, Coffee2theorems, share with us your "It is unsuitable in many ways, as discussed on talk..." reasons for reverting Carlton's text and tree. Other than that you still don't understand it, if you don't mind. Glkanter (talk) 00:26, 1 July 2010 (UTC)

Kmhkmh, why different rules for different editors? Nijdam puts into the article whatever he feels like saying. No discussion, so sources, no consensus. Is that whole Bayes discussion that's going on going to result in an edit to the article based on sources, or the best answer the meeting of the minds here can come up with? FA? That means the footnotes are in order. More BS reasons for reverting. None having to do with the merits of Carlton's solution. Glkanter (talk) 16:36, 1 July 2010 (UTC)

@Glkanter - your continued addition of this image has reached the point of disruption. You are carefully avoiding making 3 reverts within 24 hours but 2 reverts every 24 hours will also get you blocked. You really, really need to stop. -- Rick Block (talk) 04:14, 2 July 2010 (UTC)

That's weak, Glopk. All you got for reverting is OR? That's not even a fact. Whatever happened to Truth? Glkanter (talk) 15:42, 2 July 2010 (UTC)

Besides, Glopk, that's a double standard for editors. Look at that conversation about Math Formulation. Whatever comes out of there and makes it into the article will be OR. Paraphrasing text with text is OK. Why not paraphrasing text with a visual aid? Everything in that tree comes from Carlton, Whitaker, or a simple math calculation. There's no OR. Just a 'formal' statement of Carlton's (and others') solution.

This canard has already been answered. Carlton's paper already has a figure, which happens to be identical to the left-hand side of one already in the article. Yours doesn't, you invented it, it is not a simple transformation of Carlton's e.g. like replacing a font, changing a color or the thickness of a line, or adding an explanatory note. glopk (talk) 20:53, 2 July 2010 (UTC)

I'm not engaged in any canard. Saying that the decision tree is OR is a judgment on your part. Just like my opinion that the discussion you started, about "Math formulation in odds form." is no longer based on any sources, but rather a bunch of editors deciding what's 'right'. Or Rick's claim that his redundant image for the conditional solution is not OR. Opinions, judgments, interpretations all. To the best of my knowledge, Rick has just about never been right about anything related to the MHP or Wikipedia policies, so I will discount his opinion on Carlton's decision tree. Coffee2theorems still hasn't demonstrated that he understands Carlton's decision tree, so I'll discount his opinion, too.

Everything in that tree comes from Carlton, Whitaker, or a simple math calculation. And a decision tree is apparently the tool you guys prefer for modeling the problem. All I did was take existing elements and present them in a way the editors are comfortable with, and that readers will understand.

The text paragraph I keep editing back in is a 4 sentence direct quotation from Carlton's paper. Why some unknown editor's muddled, abbreviated interpretation is preferable is beyond me. 4 short sentences. Surely the article has room enough for that.

So, that's the meta discussion about that decision tree. Now lets talk about the data and results in the tree itself. It shows that some, Carlton's certainly, simple solutions are not based only on intuition, but on science. Many editors and reliable sources are confused about this, and some editors want to edit the article based on that incorrect POV. How can bringing the reader and editors clarity on such an important part of the MHP be a bad thing? Glkanter (talk) 06:53, 3 July 2010 (UTC)

Dude, what part do you not undertand of "Having a discussion about a proposed change on the talk page is NOT the same as editing the FA-class article against editorial consensus, and continuing to do so after being reverted multiple times?" Please stop harping on the Math Formulation, it doesn't even belong in the conceptual universe. Yes, calling something OR is a judgment call: and I made it, and 4 other editors have made it, and your "No true Scotsman" screeds agains Rick Block and CoffeeToTheorems do not add to your credibility. You are out-voted in a matter of judgment, so you have a choice of three: (1) You may accept the vote of the majority, take your lumps and find something else to edit. (2) You may try to argue your case again in the Talk Page, hoping to reach an editorial consensus in your favor. (3) You may continue trying to shove your edits down the collective throat, and getting reverted over and over. Guess which choice is the least likely to get you anywhere?

On the matter of the quotation versus a paraphrasi, again, I prefer the latter - an encyclopedic article need not be a collection of quotes to be accurate.glopk (talk) 17:45, 3 July 2010 (UTC)

In some cases I would agree about quoting vs paraphrasing. But, since some editors of the article insist on criticizing the simple solutions in an NPOV-violating manner, the simple solutions should at least be presented, to a reasonable extent, without editing. 4 sentences is NOT unreasonable. Glkanter (talk) 18:08, 3 July 2010 (UTC)

Finally! An acknowledgment: "You are out-voted in a matter of judgment..." You, Rick Block and Coffee2theorems? I like my odds. Glkanter (talk) 18:08, 3 July 2010 (UTC)

And there's no 'voting'. Perhaps you should brush up on what a 'consensus' means per Wikipedia. Glkanter (talk) 18:12, 3 July 2010 (UTC)

Why won't you discuss this on this appropriate talk page, as your revert of my edit is supposed to indicate your willingness to do? Glkanter (talk) 15:48, 2 July 2010 (UTC)

More than willingness, eagerness, I am literally dying to discuss with you, can't think of any other way to use my time, other than engaging in a fruitful and constructive argument with GLkanter about the fine points of the MHP. glopk (talk) 20:53, 2 July 2010 (UTC)

I know. I feel your pain. Sadly, Wikipedia requires discussion among editors to make it work. If you're not interested, I understand. But then I would anticipate you would stop reverting my edits if you're not willing to discuss it. Glkanter (talk) 22:40, 2 July 2010 (UTC)

And your response to my comment above about why your figure has no place in the article is...? glopk (talk) 23:55, 2 July 2010 (UTC)

Confusion, confusion, confusion

I guess during the next ten years there still will be confusion. I personally don't take people serious who assume the player is offered to switch before the goat door has been opened. On my part there is no confusion about this. As it says in the start of the article: the MHP is a probability puzzle. No confusion for me there. It may be of interest to treat is as a game theoretic, a psychological, a sociodramatic, or whatever problem, the main focus should be on probability. As a probability puzzle the solution has to be based on the conditional or if you like posterior probability given the opened door. No confusion for me there. I personally don't take people serious who think this may be circumvented. To take the right decision the conditional probability given the opened door has to be calculated, either by straightforward using Bayes' theorem (in its simplest form) or by using the symmetry. No confusion for me there. Anyone still confused? Nijdam (talk) 09:37, 1 July 2010 (UTC)

Wikipedia doesn't give a shit who you do or don't take seriously. Neither do I. Reliably published sources matter. They say various things, unfortunately. They're all 'entitled' to be in the article, without condescension. Even Morgan. Get off your goddam high horse and contribute towards a consensus, or tag off. Glkanter (talk) 11:28, 1 July 2010 (UTC)

Well, frankly, you're one of them. Nijdam (talk) 12:01, 1 July 2010 (UTC)

Likewise. But since we're both here, let's talk about Carlton's tree diagram, and how it refutes you're '1/3 <> 1/3' argument. Which is really all that you've brought to these pages for some time now. Glkanter (talk) 13:03, 1 July 2010 (UTC)

Nijdam, I agree, the MHP is a probability puzzle. It is not an undergraduate probability exam question. In probability puzzles it is traditional make whatever assumptions are necessary to make the solution as simple as possible.

The first problem statement was by Selvin, who first described the puzzle and called it the Monty Hall problem. He, in his second letter, made clear that the car was originally placed uniformly at random and that the host chose a goat door uniformly at random. This has the effect of making solution of the conditional problem have the same numerical answer as the unconditional problem and of making the distinction between the two cases somewhat pedantic due to an obvious symmetry.

Here you're mistaken. The symmetry does not make the problem unconditional. I told you so. It only makes it possible to calculate the conditional probability without the use of Bayes' theorem. Nijdam (talk) 16:38, 1 July 2010 (UTC)

I think symmetry justifies the use of a simple solution.Martin Hogbin (talk) 17:41, 1 July 2010 (UTC)

The most well know problem description, a letter from a member of the public to a general interest magazine, leaves so much unsaid that the problem cannot be solved without some additional assumptions. With the most common assumptions the formulation becomes identical to Selvin's, but is far from certain that this is exactly what Whitaker wanted to know. It is very likely that Whitaker did not make the distinction between the prior and posterior probabilities.

Well, we do not know what Whitaker wanted, but I'm also not interested.Nijdam (talk) 16:38, 1 July 2010 (UTC)

On what basis then do you decide what the exact problem is?

What is absolutely clear is that all the argument that made the problem so notable was about the fact that the answer is 2/3 and not 1/2; it was not about the door that the host opened.

Don't know what you're aiming at, but the surprising 2/3 instead of 1/2 is mainly due to seeing two closed doors and one open showing a goat. Nijdam (talk) 16:38, 1 July 2010 (UTC)

Exactly right, 'two closed doors and one open showing a goat'. Not 'Doors 1 and 2 closed and door 3 open showing a goat'. The numbers were added by vos Savant for clarity. She now admits that was a mistake. Martin Hogbin (talk) 17:41, 1 July 2010 (UTC)

Which numbers then? Nijdam (talk) 18:59, 1 July 2010 (UTC)

No numbers. The problem formulation should just state that an unchosen door has been opened to reveal a goat. Martin Hogbin (talk) 22:04, 1 July 2010 (UTC)

That doesn't change anything. See e.g. this and this, which state the problem without door numbers, and still give only the usual conditional solution. Because the doors are distinguishable to the player, he (i.e. you) can number them in any arbitrary way in the solution without loss of generality, even if they were not numbered in the problem statement. -- Coffee2theorems (talk) 22:22, 1 July 2010 (UTC)

Of course it changes the problem, unless you take the liberty of assuming things not stated in the problem statement, like that the player really knows which door has been opened. How do you know that? Nijdam's statement simply says, 'two closed doors and one open showing a goat'. The door opened by the host is not specified. By inconsistent problem formulation it is possible to turn the MHP into an interesting problem for probability students but it is better to be open and clear about this. Conjuring tricks do not help people to understand probability. Martin Hogbin (talk) 09:05, 2 July 2010 (UTC)

Suppose you are on the show, haven't picked your door yet, and want to solve the MHP (stated without door numbers) at that point. You can point your finger at each of the doors in any arbitrary order you wish, so suppose you do that as a part of your solution method (perhaps only mentally, so Monty won't be confused). Define "door 1" to mean the door you first pointed at, "door 2" to mean the second door you pointed at, and "door 3" to mean the remaining door. If you say that this is impossible for you, then pray tell, how is it possible for you to make your initial choice of a door either? Of course, the mere act of naming the doors changes nothing.

Eventually you will pick some door (number S), the host will open some door (number H), and there's one remaining door (number R). Eventually, you will be called on to make the switch/stay decision, and of course you can tell at that point what numbers S and H are, because you know the numbering (you assigned it yourself for convenience!) and you can observe which respective doors you and the host picked. If you are e.g. blind, I'm sure Monty will help you out here in the interests of equal opportunities. But really, the problem statement does not state anything about you being blind, and only smart-asses "solve" problems by failing to assume that the person described in it can see. So, you eventually will make your decision based on the value of P(C=R | H, S), whatever H and S will turn out to be. All you need to do now is to show that P(C=R | H, S) = 2/3, whatever the values of H and S.

Now, as a proof technique, you can assume without loss of generality that S = 1 and H = 3, because the proof is invariant to relabeling of the doors (as it must be - if it depended on the arbitrary door numbers, then arguably that's the same as saying there is no solution). But if you have doubts about such a proof technique, you don't have to use it! You can show that P(C=R | H, S) = 2/3 by laboriously calculating it for every possible door numbering. If you do that, there are no "specific door numbers", only "an arbitrary door numbering". Of course your choice of proof technique does not change the theorem. -- Coffee2theorems (talk) 12:17, 2 July 2010 (UTC)

That result is indeed correct, whatever set of host preferences you choose. Thus, in the general case, the answer is always 2/3 and the host preference is an irrelevance.

Before you start to answer any probability problem you need to decide on the basis that wish treat it. The basis suggested by most sources that support a conditional solution (such as Morgan) is that the problem must be solved by using only information given in the problem statement. On this basis, if the door opened by the host is not identified in the problem statement (as is the case in Nijdam's statement above) you cannot use the door number (or other ID) in answering the question.

Now, if you want to step into the real world that is fine but then you need to state what real word assumptions you are making, give a rationale for each, and be consistent in the way you treat unknown information. Anything else is a conjuring trick. Martin Hogbin (talk) 12:36, 2 July 2010 (UTC)

There is no such thing as "using only information given in the problem statement" without knowing something about the nature of the real world. Sure, the rules of the game need to be described, fair enough. But things such as "when you are given a choice of three doors on a game show, that means those doors are spatially distinct, and you make the choice by e.g. pointing at them" is just plain common sense (even in all the illustrations in the article they are spatially distinguishable!). The Boy or Girl paradox doesn't say that boys and girls are distinguishable either - how can you "step out to reality" to make the assumption that most people aren't boys and girls at the same time?! Well, it's just common sense! -- Coffee2theorems (talk) 14:46, 2 July 2010 (UTC)

Most of the sources that support the conditional solution warn us not to use information not given in the problem statement (I guess we all have to know what a door is or all probability questions would be meaningless) but if the problem statement does not tell us which door has been opened then it would seem to me that assuming a specific door has been opened is clearly contrary to that advice.

Of course, I have no objection to common sense either. Common sense tells us that the player in a game show would have no idea of any host preference so the door opened by the host has no meaning to the player and thus would make no difference. The simple solution is therefore correct by applying common sense.

Unless the player has watched all the earlier shows and is trying to outwit the TV company,who in turn are trying to outwit the player, in which case you want the game theory solution.

In short, there are several valid ways of addressing the problem but they all give the answer that the probability of winning by switching is 2/3. Do you not agree? Martin Hogbin (talk) 15:52, 2 July 2010 (UTC)

(undent) @Martin My previous explanation does not assume that "a specific door" has been opened, only that the player knows which door she picked and which door the host opened when she's making her switch/stay decision. In other words, it is about the probability P(C = R | H, S) for unspecified (i.e. any) H and S (R is the remaining door). That's what I think the MHP is about, and what the common conditional solutions to it are about. The MHP is not about P(C ≠ S | S), because that would assume that the player knows only the door she selected, but not the door the host opened, and that would be contrived.

It's true that the conditional solution ("mathematical formulation") in the article only concludes that P(C = 2 | H=3, S=1) = 2/3. Likewise, vos Savant's solution only concludes that P(C ≠ S | S=1) = 2/3. Both of these solutions assume a specific numbering - vos Savant's solution assumes that the player selects door 1, and the conditional solution further assumes that the host opens door 3. Neither is intended to mean that there is anything special about these numbers. Vos Savant's solution could have a similar table for every value of S in {1, 2, 3}, to prove the same thing for any S. Likewise, equations for all the numberings (H, S) in {(1, 2), (1, 3), (2, 1), (2, 3), (3, 1), (3, 2)} could be given in the conditional solution section, to prove the same thing for every (H, S) pair. Neither modification is necessary. It is obvious that any numbering will result in the same answer because of the symmetries in the solutions. It is also obvious at the outset that if there is to be a definite answer at all, then the solution will have to be invariant to renumbering of the doors.

The ways of addressing the problem that say a definite probability answer exists say that it is 2/3, i.e. P(C = R | H, S) = 2/3. There are ways of addressing the problem that say no definite probability answer exists, but nevertheless it can be said that the player should switch, i.e. P(C = R | H, S) ≥ 1/2. In a non-Bayesian setting, making the first claim requires the assumption that the host chooses randomly among the doors he's allowed to choose. -- Coffee2theorems (talk) 22:01, 3 July 2010 (UTC)

We were not discussing your interpretation of the problem we were talking about a formulation in which Nijdam specified 'two closed doors and one open showing a goat'. Note that in this formulation, the door opened by the host is not specified or identified. You may think that the player could see which door was opened but this information is not given to us. On the basis that all we know is that there are 'two closed doors and one open showing a goat' the unconditional solution applies and the answer is 2/3. Martin Hogbin (talk) 22:53, 3 July 2010 (UTC)

If we cannot assume whether or not the player can see which door is open, then neither the conditional nor the unconditional solution applies, because the former assumes the player can see, and the latter assumes the player cannot see. There is no inference without assumptions. Besides, Nijdam said "seeing two closed doors and one open showing a goat". Note the word "seeing". Even the word "showing" alone would be suggestive enough. -- Coffee2theorems (talk) 01:25, 4 July 2010 (UTC)

Well I guess I have to accept that you are correct. Not because of any of your arguments but because in statistical questions we must attempt to address the question or situation intended by the questioner. In this case it was Nijdam who made the statement and I therefore suspect that it was his intention that the door opened by the host should be identified, despite what he actually wrote. Martin Hogbin (talk) 09:17, 4 July 2010 (UTC)

Confusion: The exact meaning of "conditional, because door X has been opened" in the article is treated in some nebulous way. In case of an assumed "randomness" in the host's opening one door (if he has the choice he always will open one door "uniformly at random") there isn't given any relevant new information about the probability of the door selected by the player (1/3). No new relevant information means "conditional" without sufficient "condition". Disturbing and confusing.

But in case of any assumed "door-preference" or "extreme door-preference" of the host, relating info would surely be revealed: In opening his preferred door he is showing that the probability of the door selected by the guest has augmented to 1/2 e.g., and in opening of his "avoided door" he would clearly be showing that his preferred door actually hides the prize and the probability of the door selected by the guest has downsized to 0, e.g. So "opening of one specific door" can have two different meanings, two significantly different effects.

This difference should be emphasized. It should clearly be shown that "conditional" is not related to just opening one specific door showing a goat, but that "conditional" is related to a possibly assumed "door-preference of the host in opening one door.": Then, and only then, mentioning "conditional" would make sense. Only in treating the assumption of a possible door-preference. Without mentioning of such expressive "condition" (door-preference, and not just opening one door) the "conditional-way" is rather disturbing. Regards, --Gerhardvalentin (talk) 12:22, 4 July 2010 (UTC)

Gerhard, I have tried to find out exactly what constitutes a condition in a probability question but no one has yet told me. No doubt that statement will result in howls of protest from those who claim to have given explanations. The problem is that none of the explanations of what exactly constitutes a condition stands up to scrutiny. In the end I can only conclude that what exactly is a condition in a probability problem is somewhat arbitrary; a condition is is any event that might, in the opinion of the person answering the question, affect the probability of interest.

The point I was discussing with Coffee2theorems was concerning a comment by Nijdam. I conceded because the exact meaning of the statement should be what Nijdam intended. Others should take the same approach. Martin Hogbin (talk) 13:11, 4 July 2010 (UTC)

Please accept my apologies for having intruded in this discussion concerning Nijdam, I just used the label "confusion, confusion ..." that was welcome to me to underline my concern. - Rick wrote above about "reasoning that the host opening a door doesn't change the original 1/3 because ... and about Falk, about erroneous reasoning, the correct numeric answer for the standard version of the problem... and that it fails for slight variants where this reasoning is not sound." And I answered him that he referred to those essential aspects in a clear and concise form, and that it was fine if the article could show those aspects in a clear and coherent manner.

So I'm just underlining that the article really suffers from confusing "semi-hidden assumptions" especially regarding the "conditional solution" and that I claim for clarity and for sound arguments when talking about any "conditional solution", admitting that the "condition" always is any assumed behaviour of the host in opening a door, revealing additional hints on the actual situation, and never any meaningless "opening" only that lacks any additional information about the probability of the door selected by the player (no additional hints, no new condition). This difference has to be made clear and obvious in the whole article, from the beginning to the end, just to help understanding and to avoid any objectionable confusion. Please can you help to achieve acceptance for this aim, to avoid confusing "conditional without sufficient condition". Regards, --Gerhardvalentin (talk) 14:29, 4 July 2010 (UTC)

I agree with your comments 100%. I have been trying to accomplish exactly what you describe for nearly 2 years. Martin has been at it a little longer than I. And you can see how difficult it is to make progress. Your continued involvement and contributions will be very helpful in attaining our goal for the article. Glkanter (talk) 14:42, 4 July 2010 (UTC)

There is therefore justification for treating this problem in many ways, from the simple and notable (but undefined as either unconditional or symmetrical conditional) problem, through the interesting (conditional)undergraduate question in which the host goat door choice might matter, to the game theoretical approach in which the player and the host are regarded as adversaries. That is exactly what we do here. We start with the simple (a little vaguely defined to avoid initial complication) the we have an explanation of why the problem can be regarded as conditional followed by the conditional solution, then we should have the game theoretical solution, if for no other reason than to show that Seymann was correct when he said we must ask about the intent of the questioner to give a correct solution. Martin Hogbin (talk) 15:41, 1 July 2010 (UTC)

Many variants of the problem may be treated, as long as the formulation with the opened door is not considered to be solved by the simple solution. Let's not mix things up. Nijdam (talk) 16:38, 1 July 2010 (UTC)

I think it is perfectly justified to give a simple solution to the symmetrical conditional problem. It gives the right answer, and not just by chance. Martin Hogbin (talk) 17:36, 1 July 2010 (UTC)

Are Selvin's Problem and Whitaker's Problem BOTH The MHP?

Selvin:

3 boxes

1 set of keys, period

Contestant selects Box B

Host reveals Box A

Whitaker:

3 Doors

1 car, 2 goats

Contestant selects Door 1

Host reveals Door 3

Glkanter (talk) 06:18, 2 July 2010 (UTC)

There is not one and only one MHP. For each particular wikipedia editor, they may well be a "personal all time favourite" but no one can claim that one of the formulations is THE formulation. Not even @Nijdam. Just because certain formulations are common in probability classes does not mean that they are the unique best official formulation. Other formulations are common in economics classes. Others in psychology classes. MHP is a topic, not a standard Probability 101 problem, nor a standard Game Theory 101 . It is part of modern culture. It grows and changes. Moreover, we now know you must distinguish between Whitaker who didn't mention door numbers at all but just talks like you about "you choose a door, the host opens another, should you switch?" and vos Savant who added the semantically ambiguous "say Door 1" and "say Door 3". Did these doors already have these names or is she naming them for our convenience retrospectively to the choices of player and host? ie player picks a door (could be left, middle or right). vos Savant paints a big Number 1 on it. Host opens a door. (could be left middle or right but different from player's choice) and asks player if he would like to switch to the third door. vos Savant paints a big numeral 2 on the third door and a big numeral 3 on the opened door. (vos Savant herself explicitly stated that she intended this "retrospective naming" and that corresponds to her solution and makes it complete, and it corresponds to your solution and makes it complete.)

So my answer to your question is that both persons words are passages in the history of the MHP. Neither is "THE" MHP. There does not exist a "THE" MHP. Gill110951 (talk) 08:37, 2 July 2010 (UTC)

I obviously think Selvin and Whitaker have asked the same question (as summarized by K & W's premises). That's why the insistence that 'Door 3 be opened' is somehow unique is a fairytale in my opinion. Wasn't that nice of me to not call it BS? As far as I know, only Morgan and his crew tried to change the question, or perhaps more correctly, the premises. I agree, the solutions are limitless, but not all are created equal. I claim any solution that relies on creating a host bias (even 50/50) is a lesser solution that one that doesn't, for example, Carlton's. That includes the traditional conditional solution and the Bayes solution. Glkanter (talk) 10:32, 2 July 2010 (UTC)

I would say that Selvin and vos Savant both asked the same question. Whitaker didn't mention any door numbers and didn't mention Selvin either. I don't know if vos Savant knew about Selvin. I think "everyone" knew something about the Monty Hall show back in those days. Selvin is a biostatistician, and he proves himself not to be a very good mathematician in the more narrow or formal sense, but that doesn't matter. (Mormon et al. similarly). Vos Savant was a lady with a fantastic IQ but I have no idea what kind of training she had if any. My refined solution (showing that you can't do better than winning 2/3 of the time) takes account of possible host bias. I don't introduce it explicitly, and I don't have to. I strengthen the simple solution by proving with completely elementary arguments that not only is "always switching" much better than "always staying", but moreover there is nothing better still. The mathematicians under us realise that this is mathematically equivalent to proving all conditional probabilities are at least 1/2. Don't you see that my conclusion is an even stronger conclusion? And I didn't make any further assumptions whatsoever. There could be host bias, so if someone says it might be there, I've preempted then; if it never occurs to you or for some philosophical reason (I suspect connected with the way you think about probability) you don't believe in it, well that's your problem. Gill110951 (talk) 14:09, 3 July 2010 (UTC)

I would say that from the contestant's state of knowledge on a game show, there can be no awareness of a host bias. Of course, with 'indifference' or whatever, the contestant generalizes this to 50/50, if he thinks about it at all. But it needn't be a separately stated premise. And that the traditional conditional and Bayes solutions require this premise makes them 'restricted' solutions for those who argue that there *can* be a host bias. For those of us who said it was 50/50 all along, without an extra premise, it makes no difference whatsoever.

So, I expect the Morganians to claim 'indifference' to defend their defenseless position. Quite the turn of events. Glkanter (talk) 16:13, 3 July 2010 (UTC)

It's a puzzle about a game show. The contestant knows nothing beyond what's in the narrative. A lot of your game theory stuff is superfluous. It adds no more value to the average pub patron settling a bet regarding the Selvin/Whitaker/vos Savant MHP than the stuff raised by those 4 guys from ODU. I'll discuss probability with you, and share my thoughts about what I do or don't think. In an appropriate forum. But this is the MHP talk page. And it's 2/3 I'll choose a goat, so I should switch. Glkanter (talk) 17:30, 3 July 2010 (UTC)

Carlton

Before it is reverted again, might we have a discussion of what exactly is wring with Glkanter's edit. Martin Hogbin (talk) 11:25, 4 July 2010 (UTC)

I see I am too late. Kmhkmh, what exactly is wrong with Glkanter's version? Martin Hogbin (talk) 11:28, 4 July 2010 (UTC)

See the discussion further up with glopk and Rick. Imho glkanter should refrain from readding this, until the issue is resolved first or at least other editors have issued adiitional supportive opinions. Boldly adding material in highly contested article by an individual editor against the explicit wishes of several other editors is something that should not be done and I assume glkanter is very well aware of that.--Kmhkmh (talk) 11:42, 4 July 2010 (UTC)

What is wrong with the discussion in the section Carlton's Decision Tree? I don't see any point in copy-pasting the contents of that section here, but that seems to be what you are asking for. -- Coffee2theorems (talk) 12:03, 4 July 2010 (UTC)

I was hoping for a higher standard of discussion, particularly about which version is of most use in explaining the solution to the general public. Who wants to go first? Martin Hogbin (talk) 12:43, 4 July 2010 (UTC)

My opinion is supportive of Glkanter, though I think his version of Carlton needs some polishing. Decision trees help develop probabilistic thinking. One can give a perfectly good decision tree for the so-called popular solution. That solution is a perfectly respectable (though not unique) solution to a perfectly respectable (though not unique) mathematization of the informal question posed in vos Savant's article, for instance. The MHP is partly such a famous problem because of the richness of thought which it has engendered. The wikipedia page should give an attractive overview, make the maths and logic transparent, bring out the unity of the different approaches, and not dogmatically promote one particular POV. Gill110951 (talk) 17:23, 4 July 2010 (UTC)

So where is the case for the reverters? In what way is Glkanter's version less useful to an average reader? Martin Hogbin (talk) 22:14, 4 July 2010 (UTC)

Glkanter's diagram introduces prior and posterior probabilities, and names the doors - which is not what the cited source does. It's WP:OR as explained numerous times above. -- Rick Block (talk) 22:41, 7 July 2010 (UTC)

Who said it?

See if you can guess who answered, because you already know who asked...

"Is the extended Bayesian analysis recently added from some particular source? If so, can someone please add a reference? The article is currently undergong a featured article review, which includes making sure that everything that should be, is referenced. I fear some folks might view this extended analysis as approaching original research, which is prohibited (see WP:NOR)."

"No particular source, and definitely not original research. It is a very straightforward application of basic probability theorems and concepts (Bayes theorem, marginals and conditionals, normalization). ... I don't think any references are necessary other than the Wikipedia articles already linked to - it's really plain algebra."

Glkanter 08:00, 4 July 2010 (UTC)

@Glkanter, please give me some illumination, and also tell us not only who said these things but when. BTW I think it would be better for the wikipedia users of the world if we the editors did some more constructive and collaborative re-thinking in order to refreshen and streamline the article and add to its value and to bring it up to date, even if thereby it could not be a FA for a year or two till the dust has settled. Much better than creaking two millimeters forwards, one and a half back, endlessly, in a state of cold war ... which means freezing the content by more and more outdated and irrelevant principles. Gill110951 (talk) 17:34, 4 July 2010 (UTC)

"editors work together to arrive at an article that fairly represents current expert opinion on the subject". Please note the words "current" and "expert". Recall that verifiability and neutral point of view are equally important to the "no own reseach" principle. "As a wiki, articles are never considered complete and may be continually edited and improved. Over time, this generally results in an upward trend of quality and a growing consensus over a neutral representation of information." Gill110951 (talk) 17:49, 4 July 2010 (UTC)

Here's how I've suggested we structure the article

Those quotes are from this talk page, in 2007, as it says, during the 2nd (?) FA review. Here's a hint. The same person recently wrote this as an edit summary:

"Nice picture you got there, too bad it's not in Carlton's 2005 paper. OR - revert yet again."

Glkanter (talk) 18:44, 4 July 2010 (UTC)

Glkanter, you are beyond salvation. Yes, you miserable fixated slug, you got me, that answer was mine. It was my FIRST non-trivial contribution to Wikipedia, and I had not yet digested well, or at all, how serious a business it was to edit WP articles. But I was lucky, Rich Bloch and others showed me how it was done and, with some gentle nudge here and there I learned. The subsequent record of the discussion page and edits show that I re-worked the Bayesian analysis section so to be properly sourced (to Jeff Gill's textbook still appearing in the References). And that section passed the FAR twice along with the rest of the article. Despite you.

But of course none of that is of interest to you, oh no indeed, all you care about is what you just know to be right about the MHP. And that's it, that's the sum total of your accomplishments.

And on the basis of this strength, you show yet again what hard work is to run a community-managed and operated resource like Wikipedia. Where one single destructive personality like yourself, when endowed with vast amounts of time to waste, is enough to negate the positive contributions of half a dozen volunteers or more. Dude, you managed to anger and drive away even dicklyon. Dick frigging Lyon!!! A guy whose intellectual contributions to the progress of humanity can actually be seen and touched in revolutionary products brought to market, and read about in stacks of referred journal papers and patents. A guy who has forgotten more stochastic signal analysis than you've ever dreamed of, you twenty-year-of-analyst-work that "knows" the "right" meaning/interpretation/solution of a trivial probability puzzle. In any sane universe having a conversation with Dick Lyon would be a pleasure for anyone, and what a privilege it is that WP makes it possible. But you are insane, and am sure you regard your fight with him as some of your finest hours.

Really, dude, get yourself a life, a bong, an intern to blow you some, whatever. Just stop doing this. Just stop, stop ruining Wikipedia, stop driving away in desperation editors infinitely more valuable than yourself.

Who ARE you? Who the fuck spends a weekend culling the archived talk pages of two years ago, looking for a tidbit to quote out of context against another editor?

Everybody, please, what the heck do I need to do to help get Glkanter permanently banned? This guy needs professional help for his delusions, and his editing of WP is obviously NOT helping anyone (himself included). glopk (talk) 18:53, 4 July 2010 (UTC)

Right. Little ol' Glkanter chased away big bad Dicklyon. That statement, and you are a joke. Read my talk page. It's all there. I didn't have to read any archives. The link to that diff I quoted is on your own talk page. You haven't learned shit. You're doing THE EXACT SAME THING AGAIN with the "Math formulation in odds form" section of this talk page. Have a nice day, Professor. Glkanter (talk) 19:05, 4 July 2010 (UTC)

@Glopk, please no personal attacks. It's the deplorable condition of the article, so please help to improve it and please avoid deplorable assaults in its Talk Page. Regards, --Gerhardvalentin (talk) 19:12, 4 July 2010 (UTC)

@Gerhardvalentin, What the heck are you talking about? What No personal attacks? Of course it is personal: even GLkanter is a person, and his quoting me two years later out of context to attack me is personal. His abusive behavior personally offends the people he calls "coward" and more, especially when those people are the saint volunteers that keep this WP thing running. His disruptive behavior personally offends those of us who'd rather spend our lives on something more entertaining than avoid an FA article from being defaced. When he gets into personal fights with luminaries (e.g Dick Lyon), he damages the whole of Wikipedia - which needs those luminaries a lot more than Glkanter's. Gerhard, please, get your priorities straight: we are not writing an article about the appropriate rules of civility for the local bocce-ball club. Rather, we are dealing with a paranoic nitwit and one-article groupie that won't let go of his delusions. You can't have too many of them AND have a functional community site. glopk (talk) 20:30, 4 July 2010 (UTC)

Thank you, Glopk, for your statement. Paranoic nitwit and one-article groupie? I hope that concentrating on arguments (others than just blowing them away) could help to improve the article and could be conducive and helpful to get over the hazy state of the article. Concerning your pain: btw, in en.wp, for me, MHP is my main interest, too, hope you can forgive. Regards, --Gerhardvalentin (talk) 21:12, 4 July 2010 (UTC)

Glopk, you've already said more than once that I took your comment out of context. That is not true. There is nothing ambiguous about Rick's question about OR and your response here. Like I said earlier, I got this diff off your own talk page, Glopk. Dicklyon acted like a bully towards me and my talk page edits, despite my 3 requests that he 'leave me alone', and then tap-danced and told many different versions of what happened. There's an RfC that he and Rick filed on me. It was total BS. Dicklyon acted in an intellectually dishonest manner. No amount of 'good works' elsewhere ever excuses that. Glkanter (talk) 20:59, 4 July 2010 (UTC)

Note for the reader of a distant future. The above comment was written on a computer likely equipped with an optical mouse, and perhaps a digital camera. It refers to a wiki-row between Glkanter, a nitwit who's unaware of the meaning of the word context, and the genius who invented said optical mouse, as well as many other interesting and valuable things, including important pieces of said digital camera. Luckily for the fate of humanity, the genius quickly moved on to more interesting endeavors, whereas the nitwit kept doing his lowly thing, that is, incessantly gnaw and gnaw about the Wikipedia article concerning the Monty Hall Problem. glopk (talk) 06:48, 5 July 2010 (UTC)

Another note for the reader of a distant future. I don't know why were discussing him, but this is from The Saint's user page... Plus there's great stuff on my talk page...

Care to talk about the math, Glopk? Or just personal attacks? And based solely on this article and talk page, I think the world of Wikipedia will continue even if you and your ilk took your ball and went home. Glkanter (talk) 07:10, 5 July 2010 (UTC)

Rick Block Reported Me Again

http://en.wiki.x.io/wiki/Wikipedia:Administrators%27_noticeboard/Edit_warring#User:Glkanter_reported_by_User:Rick_Block_.28Result:_.29

Glkanter (talk) 19:25, 4 July 2010 (UTC)

11 (or more) Additional MHP Articles On Wikipedia

Selvin's problem is about the contestant choosing box B, and the host revealing an empty box A.

Whitaker's problem is about the contestant choosing door 1, and the host revealing a goat behind door 3.

Since Whitaker's question is considered 'unique' by some editors, it follows that the other 5 permutations must also be 'unique'. Certainly they are worthy of their own articles. So that's 6 articles.

Add 6 more for Selvin's problem and permutations, and we're up to 12.

Maybe another editor knows of other unique MHPs that are reliably sourced? That would be 18 Wikipedia articles, or more.

Plus, this would be a good work-around for the currently blocked article. Glkanter (talk) 03:51, 5 July 2010 (UTC)

Addition to reference list, useful reading

Since editing is currently blocked here I write down here a double literature reference which should be uncontroversial, as well as a link so everyone can read the paper in question.

M. Hogbin and W. Nijdam (2010), Comment on "Let’s Make a Deal: The Player’s Dilemma" by Morgan, Chaganty, Dahiya and Doviak. The American Statistician, Vol. 64, 193.

P. Morgan, N.R. Chaganty, R.C. Dahiya, and M.J. Doviak (2010), Response to comment by Hogbin and Nijdam. The American Statistician, Vol. 64, 193-194.

http://www.math.leidenuniv.nl/~gill/Hogbin_Nijdam_2010.pdf

Gill110951 (talk) 05:02, 5 July 2010 (UTC)

I love these guys.

"More pertinent, if the host randomly opens one of the two unselected doors (Monty does not know where the car is), then the more general problem as presented in our article shows the probability of winning by switch-ing is again 1/2. Mr.Whitaker, if after twenty years you are still interested, this is your answer". - Morgan, et al

They remain shameless. After twenty years and at least three mistakes, they still claim the unique ability to decide the one and only 'answer' to the MHP. And it's not even the MHP. It's the Random Monty variation. These guys can't be serious, can they? Glkanter (talk) 05:41, 5 July 2010 (UTC)

And how on Earth is Random Monty 'more general' than Always-A-Goat-Monty? Says who? By what definition? Glkanter (talk) 06:02, 5 July 2010 (UTC)

I think what Morgan are doing is giving an answer to Whitaker's question. In other words, if the host does not know where the goats are the answer is 1/2 but if the host does know, the answer is 2/3. More interesting is that they now agree that the answer is 2/3, regardless of host door preference or anything else. They make this perfectly clear when they say (my emphasis) 'We take this opportunity to address another issue related to our article, one that arose in vos Savant’s (1991) reply and in Bell’s (1992) letter, and has come up many times since. To wit, had we adopted conditions implicit in the problem, the answer is 2/3, period. I am not sure how this could have been put more clearly or definitely. Any solution that does not give the answer of exactly 2/3 is either wrong or answers a different question. The article should reflect this view. Martin Hogbin (talk) 10:38, 5 July 2010 (UTC)

I don't know. The title of their original paper was "Let’s Make a Deal: The Player’s Dilemma". If I can generalize that as 'The Monty Hall Problem" (and they had no idea that Whitaker was interested in the Random Monty), then, once again, they are off track. Way off track. I agree, they are giving the answer to Random Monty. And I repeat, that has nothing whatsoever to do with the MHP. But, it does give them a way to refer to their alleged more general problem and confuse things once more with an answer other than 2/3. What kind of people do things like that? Just more Morgan smoke and mirrors. You suppose Whitaker will be reading your letter and Morgan's response? Is that really the purpose for which The American Statistician gave them an opportunity to respond? Glkanter (talk) 10:57, 5 July 2010 (UTC)

Whitaker actually said, “I’ve worked out two different situations (based on Monty’s prior behavior i.e. weather or not he knows what’s behind the doors) in one situation it is to your advantage to switch, in the other there is no advantage to switch", so I think that, in their response, Morgan are just confirming that Whitaker was correct in his calculation.

Far more important is that an up-to-date, reliable, peer-reviewed source states that, with the standard rules, the answer is 2/3, regardless of anything, such as host door preference. As I have shown above, there really can be no argument that the 'Probabilistic solutions' given here are in any way superior to, or more complete than, the simple solutions. I hope that fact is becoming accepted here now. The insistence on considering which door the host might have opened is a piece of pointless pedantry.

As to the 'real MHP' all that can be said is that vos Savant's version of Whitaker's question and the simple solutions are by far the most notable. Martin Hogbin (talk) 11:15, 5 July 2010 (UTC)

The source does not say "regardless of anything, such as host door preference". What "had we adopted conditions implicit in the problem, the answer is 2/3" means is that:

• They think that the conditions that the car is randomly placed and that the host is unbiased are "unstated conditions" (their words, again) that are implied, rather than expressly stated in the problem.
• They think that adopting these conditions results in the answer 2/3 (using the conditional solution; they have not retracted any of their criticism of the unconditional solutions).

The article does reflect this, because the solutions that do not make the standard assumptions are in the variants section, not in the solutions section (although Gill added some of the q stuff to the conditional section, it is qualified by saying that it relaxes the assumptions). -- Coffee2theorems (talk) 11:47, 5 July 2010 (UTC)

To address your first point, I may have put it a little too strongly. What I mean is that Morgan have agreed that, to the question as asked, the answer is 2/3. Whether this is because the host is assumed to have chosen a goat door uniformly at random or because the host may have a preference but we should apply the principle of indifference from the player's SoK we do not know. The point is that we need not take any potential goat door preference of the host into account in our solution.

Bearing in mind the above, the fact that we know the door number opened by the host does not matter means that the argument that we must show a solution in which two possible doors might be opened by the host is pedantic to say the least. Bearing in mind also that the 'Probabilistic solutions' presented in this article are not strictly correct either, in that they do not show other doors that the player might have chosen, there is no justification for claiming that the 'Probabilistic solutions' are in any way superior to the simple solutions. Martin Hogbin (talk) 12:05, 5 July 2010 (UTC)

Actually, if the question is why are Morgan et al. saying the answer is 2/3 I think we do know, because (as Coffee2theorems says above) they said why, and since they haven't retracted anything they said about the unconditional solutions being false they're still saying they consider the problem to be inherently conditional (which they've explicitly said before, i.e. in their rejoinder to Seymann). In addition, this hasn't in the least negated any of the other numerous sources that are critical of unconditional solutions. Bearing this in mind, the article must present both types of solutions in an NPOV manner, and must also include the criticisms of the unconditional solutions. -- Rick Block (talk) 13:52, 5 July 2010 (UTC)

Page protection

Because of a minor edit war we are all to be treated as a bunch of petulant children. This is an unnecessary overreaction. For most of this two year dispute discussion here has been of a good standard, focusing on the content of the article rather than personal attacks, and not edit warring. I am shure that we could return to that standard of discussion with a little effort from all. WP is the meant to be encyclopedia that anyone can edit with articles being subject to continuous improvement. There are many different views held by the regular editors here but I do not believe that many think that the article now is as good as it can ever be.

I would therefore like to propose that we ask for the page to be unlocked so that the process of improvement can continue. With regard to the current edit war. I strongly suggest that both sides continue discussion, regardless of the fact that you may claim to have said it all before. Freezing pages achieves nothing, discussion might, if all sides show some more willingness to talk.

Would anyone else like to see this page unlocked? Martin Hogbin (talk) 10:17, 5 July 2010 (UTC)

The primary goal of WP is to be an encyclopedia not not an anyone can edit forum. Anyone can edit is merely the primary tool to achieve that goal, but just a tool and not the only one. I think the long protection might actually be a good idea, as it might help the editors here to settle differences first and focus on the main issues (in particular the long time conflict between Rick and Martin regarding the structuring/ordering of simple and conditional solutions and the larger changes suggested by gill). There should be a focus on these two and an attempt to come to an agreement here, rather than starting additional conflicts (like glkanter did) or having lots a minor edits which in doubt just add confusion and make it harder to track things.--Kmhkmh (talk) 12:06, 5 July 2010 (UTC)

For nearly all of the long term conflict all editors here exhibited exemplary behaviour. Even now the recent edit war is relatively minor and limited in scope to a few editors. I do not see why everyone should be prevented from editing. Martin Hogbin (talk) 12:58, 5 July 2010 (UTC)

I see no reason for any urgent edits, but I see a lot of reasons to resolve the points mentioned above.--Kmhkmh (talk) 13:22, 5 July 2010 (UTC)

I also see no reason for any urgent edits - and editing is not prevented, it just requires asking an admin to make edits that have been agreed to on this page - see Template:Editprotected. The point of protection in this kind of circumstance is to force users to hash things out on the talk page, rather than continuing a (seemingly endless) edit/revert cycle. -- Rick Block (talk) 14:05, 5 July 2010 (UTC)

You seem to see a need for urgent edits below. I and many others who have been prevented from editing this page were not part of any edit/revert cycle. —Preceding unsigned comment added by Martin Hogbin (talk • contribs)

Feel free to take this up with the admin who responded this way to the 3rr report. Admins are empowered to use their judgment. Rather than block a particular user in this case, the responding admin clearly felt that there was a larger problem that would be addressed by protecting the article. It's certainly not an unreasonable choice. -- Rick Block (talk) 15:23, 6 July 2010 (UTC)

Remove protection - the article is simply wrong

The cowards who have defended the false solution (which was made notorious by a so-called "gemius - Marilyn vos Savant) have no right to block the truth from being placed in the article. —Preceding unsigned comment added by 208.127.183.174 (talk) 17:24, 7 July 2010 (UTC)

pure rubbish - a kindergarten student could see this

"The critical fact is that the host does not always have a choice (whether random or not) between the two remaining doors. He always chooses a door that he knows hides a goat after the contestant has made their choice."

BULL! 1/3 of the time, the contestant chooses the door with the goat. —Preceding unsigned comment added by 208.127.183.174 (talk) 17:28, 7 July 2010 (UTC)

???  -  And in 1/3 of the time, she chooses the door with the SECOND goat. And in 1/3 she chooses the door hiding the one and only car and, in that latter 1/3 only, the host has a choice. What are you talking about? --Gerhardvalentin (talk) 17:40, 7 July 2010 (UTC)

Like all the defenders, you don't know the game (or are just lying) - only ONE door had the gag gift behind it. —Preceding unsigned comment added by 208.127.183.174 (talk) 18:10, 7 July 2010 (UTC)

Sadly, in many ways, this new editor's 'contributions' are of equal value to those I have been arguing against for nearly 2 years. Even to this day. Glkanter (talk) 18:34, 7 July 2010 (UTC)

It is the article itself that should be convincing this editor. How can we improve it so that no one can fail to understand? That is the question that we should all be asking ourselves. Martin Hogbin (talk) 19:21, 7 July 2010 (UTC)

User 208 is trolling about and no article improvement can stop that. hydnjo (talk) 00:57, 8 July 2010 (UTC)

You may well be right about this particular editor, but we get a regular stream of readers who do not understand the solutions presented here. This shows the article might still be improved. Martin Hogbin (talk) 09:29, 8 July 2010 (UTC)

I think the article is clear enough; some people simply refuse to understand. And, in fact, the article mentions that very phenomenon, so I think your bases are covered. --Tethros (talk) 20:52, 8 July 2010 (UTC)

Propose restoring Falk's second point

Just before the page was protected, Martin deleted the paragraph about "no news" intuition being the basis of solutions saying the host opening one of the two doors known to hide at least one goat does not change the probabiloity that the player's selected door hides a goat. Here's the diff [1]. I suggest we ask an admin to add this back. There clearly was not a consensus to delete this (based on prior discussion, above). Martin's claim that no one has explained what this means or it's relevance is odd, since this has been explained. It has been in the article since the last featured article review. -- Rick Block (talk) 14:14, 5 July 2010 (UTC)

Agree. glopk (talk) 17:06, 5 July 2010 (UTC)

Perhaps Glopk you could explain to me what this paragraph means and what 'confusion' it refers to.

Why insist on inpenatrable meaningless paraphrased barely-relavent gibberish in the article? Who benefits? Glkanter

Oppose Firstly this is not the way to proceed with editing a WP article, to get the article locked and then ask an admin to decide which edits are the right ones. How can an external admin be expected to evaluate the reasons for restoring this text, we have spent years discussing this subject. If, like me, you want to get on with editing this article I suggest that you ask to have it unlocked so that we can all continue to improve it.

Secondly I started a discussion on the paragraph to which no one but showed Rick support. That hardly shows a consensus to keep it. I then posted a special section which contained, 'There is no consensus as to why this paragraph is in the article is or what it even means. I propose to remove it unless somebody can give a clear indication of its exact meaning and purpose.' I got no response to this from anyone. I therefore believe there is a clear consensus to delete this text.

Finally, despite your protestations to the contrary, all you have done is to give your interpretation of what the text means. It is not clear to me how the words written mean what you claim nor their exact relation to what Falk said (maybe it is a direct quote, I do not know.) The meaning you claimed was not relevant to the section the text was in, which discusses reasons why people think the answer is 1/2 rather than 2/3. Your claimed meaning for this paragraph has nothing to do with this confusion. No one else has even tried to explain what the text means and why it belongs in that section. There is no consensus to keep it. Martin Hogbin (talk) 22:14, 5 July 2010 (UTC)

There was obviously no consent to remove that part. You and Rick in particular where arguing about that for a while already and it wasn't exactly a new addition to the article either but a new deletion by you. Please don't distort the facts and timeline here.--Kmhkmh (talk) 01:31, 6 July 2010 (UTC)

I started a discussion on the subject on 29 June and deleted the section on 4 July, shortly after Rick reverted an edit to it by someone else. I had no idea that the page would be so abruptly locked. Martin Hogbin (talk) 09:16, 6 July 2010 (UTC)

You restarted the conversation on June 29th and Rick disagreed with you immediately and pointed out that topic was already discussed earlier, where you disagreed with Rick as well (see [2], Talk:Monty_Hall_problem/Archive_16#Information.3F). In short the discussion Did not start on June 29th nor was there ever any consent for removal. I strongly suggest you drop such misleading formulations as above, if you want to be taken seriously.--Kmhkmh (talk) 11:43, 6 July 2010 (UTC)

The events are there in the record for all to see. I had no idea that the article would be so abrupty locked when I made my edit. Martin Hogbin (talk) 13:11, 6 July 2010 (UTC)

Nobody suggested you were aware of the article being protected shortly after your edit nor has that anything to do with the point I raised above. Indeed everybody can check the accuracy of the various comments easily for themselves, that's exactly why I provided the links for the convenience of others.--Kmhkmh (talk) 13:17, 6 July 2010 (UTC)

How about we talk about what Falk says and its relevance to the 'Sources of confusion' section. Saying it has all been discussed before is not good enough. Martin Hogbin (talk) 14:41, 6 July 2010 (UTC)

Well I can't see the Falk part should be treated differently from the rest. It is not of any particular importance. Work out a common version here first and add it in August, when the protection is over.--Kmhkmh (talk) 01:25, 6 July 2010 (UTC)

I agree. Let us discuss the subject. There is a section above or we can start again below. You could start by telling me what you understand the paragraph to mean. Martin Hogbin (talk) 09:16, 6 July 2010 (UTC)

Oppose, never in this confusing form. - Rick, on July 4th I said in your talk page:

Hello Rick, please consider that this is not "three prisoners" etc., but that this is the MHP. And in the MHP the simple solution (2/3 by switching) is always "correct". Just in case that additional info is revealed, the answer can become "closer", so from 2/3 to either 1/2 or to 1 (3/3), just a "closer" result. So, what the MHP is concerned, the "simple solution" can be said "not to be correct enough", never to be "not correct at all". It is just confusing and not helpful if a statement, treating quite another issue, is cited in the MHP without pointer/advice. Confusion? Please give me your view of what is helpful and what isn't. Regards, Gerhardvalentin - But you didn't even answer.

So never restore that paragraph that is not helpful in this form. The article suffers long enough from false reasoning and from confusion, unable to distinguish different presuppositions and requirements. Confusion should be avoided, not encouraged. Regards, --Gerhardvalentin (talk) 09:41, 6 July 2010 (UTC)

@Martin - The way edits work when a page is protected is requests can be made on the talk page - if there is an obvious consensus, adding {{editprotected}} to the relevant section of the talk page draws the attention of an admin willing to make edits to protected pages. The admin doesn't "evaluate the reasons" for the change but simply implements the change. This would only happen here with an obvious consensus for the change. Since there is no consensus to re-add this, that's not the case here. However, I'd like to note for the record there was no consensus to delete this either. Per Wikipedia:Requests for mediation/Open Tasks I think this will be the next case to get a mediator assigned. We can discuss this in mediation, but I think the "starting" version of the article should not be the protected one but one that includes the paragraph you deleted. Regarding what the paragraph means, how are the explanations already provided not clear? The point Falk is making is that there are two competing intuitive solutions, and neither is based on sound probabilistic reasoning. That one results in the correct numeric answer (given the standard assumptions) does not make it a sound basis for determining the conditional probability of winning the car by switching as can be demonstrated by a slight variant of the problem (where the intuition results in the wrong answer). This is actually the reason many sources contrast the standard MHP with the "host forgets" or "Monty crawls" variants where the apparently same setup results in different conditional probabilities. I don't have Falk's paper handy so can't provide an exact quote at the moment, but she says that learning that the host's action of opening a door can affect the probability that the car is behind the player's door was quite surprising to her (echoing Morgan's observation that the difference between the conditional and unconditional situations confounds many). IMO, since this is a widely used basis for simple solutions it is something that should be explicitly mentioned in the article.

Typical Rick Block BS interpretation/conclusion of Wikipedia policies or something...

"Since there is no consensus to re-add this, that's not the case here."

"...but I think the "starting" version of the article should not be the protected one but one that includes the paragraph you deleted..."

Yes, somehow, it was 'magical'. So, even without a consensus to add it back, it gets added back in Rick's world. Please, enough of this malarkey!

Say, I have an idea! Let's 'start' with a version that had Carlton's full quote, the decision tree, and that crappy text removed from the beginning of the Solution section!

Or, we could simply and obviously agree that unless an edit is going to save the world, or something equally important, there's no need to involve an admin to restore one of Rick's pet paragraphs. Glkanter (talk) 15:34, 6 July 2010 (UTC)

@Kmhkmh - I'm not particularly hopeful I can hash anything out with Martin outside of mediation. For whatever reason we generally seem to end up talking past each other.

@Gerhard - I'm sorry for not replying. I have been (and still will be for a few more days) otherwise occupied in real life and have very limited connectivity. Does my reply to Martin address your comment?

-- Rick Block (talk) 15:11, 6 July 2010 (UTC)

Thank you, Rick, for your answers here.

Your argument that the "simple solution" was "not correct at all" was incorrect and confusing. The "simple solution" (viewed from the player's knowledge) never is "not correct at all" (your words), but is always "correct". Of course, if s.o. should assume that opening of one door could be accompanied by any additional information then, at most, the answer exceptionally could become "slightly closer" by such a hint, but on a large scale never leaving the range of 2/3, so forever proving the result of the "simple solution". In order that such exceptional "closer hints" could become of any relevance however, such "closer hints" had to be visible to the guest. Otherwise this is purely an irrelevant "what-if" fiction, a reformulation of the rules for training purposes for math students only, not addressing the meaning of the vos Savant question, not addressing the MHP. Math doesn't have to "prove" anything in the MHP, a "mathematical answer" just proves that it answers the specific presuppositions of the underlying assumptions. Never more. As such "assumptions" are not implied at all in the original vos Savant question, they are quite irrelevant in the MHP, treating different variants.

You are not admitting that all "conditionals" obviously treat quite other issues, and that, remaining in the large scale of 2/3, such "what-if" fictions always prove the 2/3 answer, but that they never address the original question. This is nebulizing and confusing. You are preventing that these basics are made clear and obvious in the article. Please consider what can be helpful for the article and what isn't. Regards, --Gerhardvalentin (talk) 11:14, 7 July 2010 (UTC)

They address the "original" question as much as vos Savant does and it is not her question to begin with but whitaker's. The "nebulizing" in the articles begins, when editors attempt to "hide" the different takes on the problem in favour of their personal opinion/preference instead of sticking to the reputable sources and explaining the various subtle differences carefully to the readers. In that regard I'd like to recommend you to heed your own advice.--Kmhkmh (talk) 11:45, 7 July 2010 (UTC)

You say it, Kmhkmh. And if those "totally different assumptions" and "different takes" are not presented as what they are: deviations from the original question and pure "what-if" fictions. Making this clear would help the article, so it would be great if you help to make this clear. We've already so long been waiting for that, for such clarification. The article really needs it. Regards, --Gerhardvalentin (talk) 12:01, 7 July 2010 (UTC)

The article probably can do a better job in explaning different assumption or takes on the problem - I agree. However one big road block for achieving that are editors insisting on vos Savant's take "being the real thing" while in reality it is just one of different takes (with its own assumptions).--Kmhkmh (talk) 12:21, 7 July 2010 (UTC)

Wonderful, thank you Kmhkmh. You are right: History of the "MHP" shows various different assumptions and approaches. Often neglecting that it's the guest's view only, and her best possible knowledge that is concerned. The "simple solution being the real thing" is only valid if no additional far-fetched conjecture is drawn up, fictions that need additional presumptions. In case of including such far-fetched conjectures they have to be expressly stated and clearly named. Never any "what-if-she-knew"- and what-if-there-was"-probability-calculation without first providing for and clearly naming those presumed differences of the specific situation of / in each and every variant. It was fine if all those "what-if-she-knew"-fictions and "what-if-there-was"-fictions would clearly show their specific presumptions, settled there where they belong: In a completely separate section, and labeled as such. I strongly hope that it really can be possible to help stopping uncomfortable confusion and blending, and by that achieving a clear and open breakdown of segregation. Naming anything clearly and expressly. I hope that this is your main objective, also. Your help is needed. Thank you once more. Regards, --Gerhardvalentin (talk) 16:50, 7 July 2010 (UTC)

Kmhkmh, maybe you could share with us how the Whitaker/vos Savant version differs from Selvin's version with his 2nd letter premises? Glkanter (talk) 13:43, 7 July 2010 (UTC)

One more thing, please Kmhkmh, what other 'takes' are out there in reality? Morgans?

No, can't be Morgan. In their response to Martin and Nijdam's letter they say:

"To wit, had we adopted conditions implicit in the problem, the answer is 2/3, period."

If I may substitute 'assumptions' where they say 'conditions', they're in agreement with vos Savant. Glkanter (talk) 14:06, 7 July 2010 (UTC)

And for the record, this editor understands all 'takes' (solutions, really) from reliably published sources belong in the article. I just reject that there should be any qualifications or weasel wording representing any POVs prior to the solutions being given. I've suggested repeatedly a section to include all so-called 'controversies' that would come subsequent to the solutions. Glkanter (talk) 13:51, 7 July 2010 (UTC)

Agreed also. Contrary to what some may think I am not trying to promote any particular solution. There are many solutions, all valid under different assumptions. There is a set of assumptions (not just the unconditional case) for which the simple solution is valid, there are other (not much wider) assumptions under which the Morgan solution is valid, and there are assumptions under which neither of these is valid. One problem is that no reliable sources fully explain all this. Martin Hogbin (talk) 14:14, 7 July 2010 (UTC)

Martin, I don't completely agree with what you wrote above. And I'm willing to show my ignorance. Selvin came up with the game show problem and named it The Monty Hall Problem. He solved it with a simple solution. By his 2nd letter, he had all the premises clearly stated, and to prove his point, he solved the problem with the traditional 50/50 conditional solution. vos Savant, via Whitaker, makes the problem famous. She, too, solves it with a simple solution. She and Whitaker are nowhere nearly as rigorous with the premises as Selvin, but she argues extensively, in my words, that she was solving the same problem as Selvin. So, who else has published a 'similar' problem? Morgan? They claimed they were correcting vos Savant's version. But all they did was ignore Selvin's 50/50 premise. Then they admitted they missed it. Besides, Carlton's decision tree shows the 50/50 premise is not required. Otherwise, who else has published an MHP, and how many Johnny-come-latelys get to call their version of this puzzle the MHP, anyways? For example, with Random Monty, it's Deal or No Deal, not the MHP. Completely different problem. [This thought just occurred to me. Please tell me Rick isn't going to claim all those college text books teaching conditional probability are promoting a 'host bias' MHP. I don't think I could take it. Because it's the contestant's SoK on a game show that matters here.] Glkanter (talk) 16:52, 7 July 2010 (UTC)

I do not really disagree with most of what you say, but it is possible to contrive a formulation of the MHP to which Morgan's solution is the only correct one (see here [3]) but I agree with you that this is not a realistic formulation of any of the notable problem statements.

Carlton's solution

Regarding Carlton's tree. That is based on the assumption that the host chooses a goat door uniformly at random, it does not prove that is the case, nothing can, it is up to us to decide on what exact question we wish to answer. If we decided that we wished to answer the (unrealistic) case where the host door preference is known, then Carlton's solution would be incorrect. If, on the other hand, we make the decision to treat the host door preference as uniform at random (as specified by Selvin) then Carlton's tree is correct, except that some here might quibble that it does not show the two equally likely and irrelevant door options that the host had. Martin Hogbin (talk) 10:40, 8 July 2010 (UTC)

No, Martin, Carlton's tree, like many simple solutions is not door-centric. 50/50, 0/100, 100/0, and anything in-between all work. And that probability, unlike in the 'stopped clock' conditional and Bayes solutions, never needs to be specified. Of course, the contestant's SoK is equivalent to 50/50, so I would say those solutions are correct, solely due to indifference. But the whole 'host bias' thing is proven to be a canard, a prevarication. Glkanter (talk) 11:20, 8 July 2010 (UTC)

I agree that we should take the host's goat door choice to be uniform at random, regardless of how the host actually chooses, because the player does not know the host policy. These are the reasonable assumption to make and these assumptions are implicit in Carlton's tree. Thus Carlton's tree shows that a reliable source has made the same assumptions that you and I make but no mathematical solution can prove that those assumptions are correct. Martin Hogbin (talk) 12:32, 8 July 2010 (UTC)

No, Martin. Carlton's solution and tree are 'agnostic' in regards to how the host chooses between 2 goats. Please re-read my comment immediately above. In fact, based on the arguments from other editors, the traditional conditional and Bayes solutions give the wrong reason for why it's 50/50 (contestant's indifference is the only plausible reasoning), and are therefore 'false' or 'incomplete', despite giving the correct numeric answer. It's probably a source of confusion, as well. Glkanter (talk) 14:42, 8 July 2010 (UTC)

If Carlton is agnostic about how the host chooses between goat doors that simply means that he has already made the assumption that it does not matter which door the host opens. Without making that assumption you must number or otherwise identify the doors in your solution. If the host is known, when possible under the rules, to always open the door that he did not open this time then the player is certain to win by switching.

Perhaps Carlton interprets Whitaker's statement as not intending to identify doors, as mentioned here [4]. In that case his solution is fine, except that he ought to say that he considers the doors not to be distinguished in the problem statement. Martin Hogbin (talk) 15:14, 8 July 2010 (UTC)

I see 'being agnostic' about doors as being consistent with 'no new information is revealed' regardless of whether door 3 or door 2 is opened. How the host selects a door simply is immaterial to the simple solutions. That he reveals door 3 is not contradicted by this. Glkanter (talk) 16:08, 8 July 2010 (UTC)

They all mean much the same thing but they are all based on decisions to be made at the start on how to treat the problem. If we decide that any of the following apply:

1. The player has to decide before a door is opened
2. The problem does not distinguish between doors
3. The host chooses uniformly at random between goat doors *
4. The player does not know the host's door opening policy and we decide to apply the principle of indifference *

* Subject to quibbling by some here

then Carlton's solution is fine. In other words, for Carlton's solution to be correct, at least one of the above must be assumed (unless I have missed one). Martin Hogbin (talk) 16:48, 8 July 2010 (UTC)

I still disagree. None of those 4 choices are required by Carlton's simple solution. Look at the tree, please. The premise reads, 'The host will (always) open a door revealing a goat'. Gill suggests taking the door #s out of the tree. It's immaterial to me, really. Glkanter (talk) 17:46, 8 July 2010 (UTC)

Take Gill's advice, see my 2 above, then the tree will be fine but some may argue that door numbers are given in the question. It is no use looking at the tree because it does not cover the case of a known host goat door preference.

Answer me this, if the host is known, when possible under the rules, to always open the door that he did not open this time, what is the probability of winning by switching? Martin Hogbin (talk) 18:26, 8 July 2010 (UTC)

The MHP does not include "the case of a known host goat door preference". Only Morgan's (you know of them, yes?) contrivance has that. Can you re-phrase that last question? Reading it makes my head hurt. Glkanter (talk) 18:50, 8 July 2010 (UTC)

You should be used to your head hurting by now. If the MHP does not include "the case of a known host goat door preference" then it falls under my category 4 above.

I think you have got the point of my last remark anyway. It was just an example of a known host goat door preference without door numbers, where the answer was not 2/3, thus Carlton's tree would not apply. Martin Hogbin (talk) 20:29, 8 July 2010 (UTC)

Yes, 'contestant's indifference' is a valid assumption. But not needed in Carlton's solution and tree. Plus, all the criticisms of the simple solutions argue it is not a valid assumption. In that spirit, I claim those critics, who rely on the 50/50 'host bias' premise for their conditional or Bayes solutions are using the 'wrong reasoning'. You know, they get the right answer, but for an unsupportable reason, much like Falk's (via Rick) interpretation of 'intuition'. The so-called 'host bias' is immaterial, and actually, non-existent. All that matters is the contestant's SoK. Glkanter (talk) 16:59, 10 July 2010 (UTC)

Falk 3

Perhaps we can now discuss the deleted section. Here it is again.

A competing deeply rooted intuition at work in the Monty Hall problem is the belief that exposing information that is already known does not affect probabilities (Falk 1992:207). This intuition is the basis of solutions to the problem that assert the host's action of opening a door does not change the player's initial 1/3 chance of selecting the car. For the fully explicit problem this intuition leads to the correct numerical answer, 2/3 chance of winning the car by switching, but leads to the same solution for slightly modified problems where this answer is not correct (Falk 1992:207).

It was in the 'Sources of confusion' section.

Rick said, She's saying that solutions that say things like "since we already know one of the two unopened doors must be a goat, opening one and showing a goat cannot affect the probability of the originally picked door being the car" appeal to this notion, but this reasoning is no more valid than the naive "there are two doors left so they must have equal probabilities" reasoning. Both of these are based on deeply held intuitions. Neither are valid reasoning in this case.

Perhaps those who support it could answer the following:

What is the confusion that we are looking for a source for?

Most of this section deals with the fact that most people think the answer is 1/2 rather than 2/3.

Per just above: The point Falk is making is that there are two competing intuitive solutions, and neither is based on sound probabilistic reasoning. That one results in the correct numeric answer (given the standard assumptions) does not make it a sound basis for determining the conditional probability of winning the car by switching as can be demonstrated by a slight variant of the problem (where the intuition results in the wrong answer). This is actually the reason many sources contrast the standard MHP with the "host forgets" or "Monty crawls" variants where the apparently same setup results in different conditional probabilities. I don't have Falk's paper handy so can't provide an exact quote at the moment, but she says that learning that the host's action of opening a door can affect the probability that the car is behind the player's door was quite surprising to her (echoing Morgan's observation that the difference between the conditional and unconditional situations confounds many). IMO, since this is a widely used basis for simple solutions it is something that should be explicitly mentioned in the article.

The confusion is that people's intuitions lead them to a very strongly held belief in a particular answer. Most people think the answer is 1/2 rather than 2/3 because of the "equal probability" intuition. Simiarly, many people who think the answer is 2/3 think so because of the "no news" intuition and this is not based on sound reasoning either. -- Rick Block (talk) 16:12, 6 July 2010 (UTC)

Yes, this is consistent with your arguments to me for almost 2 years now that 'simple solutions fail because they do not work for this slightly different problem...' and then you give a host bias, or Random Monty/Host Forgets, or whatever. The reasoning is invalid. You've failed to acknowledge that, and continue to do so. Also, as per Carlton's tree, there's much more than 'intuition' at work with the simple solutions, there is correctness, validity, math, science, etc... But if 'intuition' means 'common sense' and 'problem solving ability', I'll gladly take a big bag of it. 'No news' given the other assumptions *is* correct. Change an assumption, maybe we're playing Deal Or No Deal instead of Let's Make A Deal. Another confusion you share with Morgan. Somehow, in their response to Nijdam and Martin, they tell Whitaker that the answer can be 50/50. Sure, on Deal Or No Deal, but not Selvin's or Whitaker's or vos Savant's MHP. Glkanter (talk) 17:32, 6 July 2010 (UTC)

@Rick Your first sentence second sentence in the last paragraph is fine, there is no doubt that the confusion most people get with the MHP is related to thinking that the answer is 1/2.

After that you are conflating two completely different issues about Monty's behaviour. The fact that Monty knows where the goats are is agreed by everyone here to be important. This, surprisingly to most people, affects the answer. It also has nothing to do with a no-news argument. If the host is known to choose an unchosen door randomly and happens to reveal a goat it does give information about what is behind the originally chosen door, there is news, and it makes the car more likely to be behind the door the player first chose. Thus intuition proves correct; information is revealed and, as a consequence, probabilities change. It is generally assumed that the player knows the game rules.

The door Monty opens, when he has a choice, is a completely different matter. The player would not be expected to know of any door preference that the host might have (and in the standard formulation he chooses uniformly at random anyway) so the particular door that the host opens gives no information to the player and does not change the probability that the originally chosen door hides the car. So in this case intuition again proves correct, no news results in no change in probability.

How does this square with what you say Falk means?Martin Hogbin (talk) 17:42, 6 July 2010 (UTC)

Martin, will this redundant madness ever end? Glkanter (talk) 18:34, 6 July 2010 (UTC)

Think about it. Rick's been repeating/flogging that erroneous (it doesn't work for slightly different problems) 'explanation' for why simple solutions are wrong for the 2 years we know about, and probably 5+ years by now. Now, if you want to ban someone from Wikipedia for Just Cause, that's the type of shit that causes untold wasted editor' hours on moronic discussions. Think of all the previous editors who eventually gave up in the face of that same incoherent argument! Glkanter (talk) 19:45, 6 July 2010 (UTC)

Other editors may have given up but I have not. Of course, we now know that the so called 'Probabilistic solutions' also fail for slightly different problems, thus they are no better than the simple solutions. Martin Hogbin (talk) 21:46, 6 July 2010 (UTC)

@Martin - I'm not conflating anything. You're refusing to listen to what the point is - which is that relying on intuition, whatever it is, is not a good way to approach probability problems. The naive "two doors, one car, therefore the probability must be 1/2" intuition fails for the standard interpretation of the problem. The more sophisticated "I already know one of the doors is a goat, so showing me a specific door is a goat doesn't change my initial probability" intuition fails for other versions of the problem. Fundamentally, she's criticizing solutions based on this intuition. -- Rick Block (talk) 22:32, 7 July 2010 (UTC)

The general statement that intuition cannot always be trusted in probability problems is fine, this problem demonstrates that perfectly, if that is all you want to say that is fine with me, However, when we get into specific areas where intuition might lead us astray you are mixing up two different areas of confusion. See new section below. Martin Hogbin (talk) 11:47, 8 July 2010 (UTC)

Well, I would phrase it a little differently. Rather than the prior allegations that the simple solutions are flawed, I prefer to say that the simple solutions, as demonstrated by Carlton's decision tree are the most elegant. The solutions that require the 50/50 host are still correct, as that is the contestant's state of knowledge. All that previous BS that Rick, Nijdam, and some others have spouted is, as we've always said, just noise to be dismissed with as little attention as possible. Wikipedia policy says some of that nonsense goes in the article. But no way should that BS dictate an errant POV in the article. Glkanter (talk) 23:34, 6 July 2010 (UTC)

Is Falk asserting that, exposing information that is already known does not affect probabilities is true or false?

She is asserting that some solutions (those that say the reason the probability of the player's door remains 1/3 after the host opens a door is because we know the host must open a door and we know one of the two doors must hide a goat) are based on this intuition and that these solutions are not based on sound probabilistic reasoning. She is not saying exposing information that is already known does not affect probabilities is true or false, just that it is a fundamental intuition (like the "two unknowns means both possibilities must be 1/2" intuition) that leads people to believe in solutions that are not mathematically sound. -- Rick Block (talk) 16:24, 6 July 2010 (UTC)

That really does not answer the question. She must either be saying that the statement is false and thus people who believe it are led astray or she must be saying that it is true but there is some other reason in this case why it still leads to a wrong conclusion. Which do you think it is? Martin Hogbin (talk) 17:18, 6 July 2010 (UTC)

How many ways do I have to say this before you understand it? She's saying relying on intuition rather than a sound probabilistic analysis (conditional, in this case) leads to solutions that are not sound. Relying on either the intuition that two unknowns must have equal probability (which is a correct intuition in some circumstances, but leads to the 1/2 answer) or the intuition that the host openining a door reveals no information because we already know one of the two doors must be a goat (which is a correct intuition in some circumstances, and leads to the 2/3 answer) leads to incorrect answers depending on the question. I.e. answers based on either of these intuitions are not sound. -- Rick Block (talk) 22:22, 7 July 2010 (UTC)

Rick, it is a simple question, which you are resolutely refusing to answer. Does exposing information that is already known affect probabilities or not? Martin Hogbin (talk) 11:50, 8 July 2010 (UTC)

The question that is the topic of this section is whether Falk asserts whether this is true or false. I've answered this question several times. She does neither. It is of course logically impossible for known information to affect probabilities. Is this what you want to hear? The summary Kmhkmh presents below is quite accurate. Her point is that knowing which door the host opens actually can convey additional information so basing a solution on the reasoning that this cannot convey additional information makes that solution based on incorrect reasoning (Morgan would say "false"). --Rick Block (talk) 19:18, 8 July 2010 (UTC)

So we agree on several things now. The intuition that no news means no change in probability is correct. The confusion being referred to here is not the reason that people think the answer is 1/2 but the fact that some people do not realise that the probability that the car is behind the originally chosen door can be changed if there is a known host goat door preference and the door opened by the host is known. Are we agreed on that? Martin Hogbin (talk) 19:34, 8 July 2010 (UTC)

More generically, the confusion is that the probabilities after the host opens a door are conditional probabilities that are different from the unconditional probabilities that are obviously 1/3:1/3:1/3 before the host opens a door, i.e. the confusion is how to figure out what the two questions marks are after the host opens a door and the probabilities are clearly ?:?:0. -- Rick Block (talk) 19:55, 8 July 2010 (UTC)

See my comment below. There are two sets of the confused, those that get the wrong answer (most people), and those who get the right answer for the 'wrong' reason ( a few). Martin Hogbin (talk) 20:01, 8 July 2010 (UTC)

Consensus to restore?

So far only one person (Rick) has even attempted to explain what the disputed paragraph means. We have not even got on to discussing whether Rick's meaning is what Falk actually meant, if either Rick's or Falk's meaning is correct, and how whatever the paragraph is eventually taken to mean mean is relevant to the section that it was in. As part of an encyclopedia for the general public this text falls at the first hurdle. Only one long-term editor of this article has even claimed to understand what the paragraph is all about. That falls very far short of a consensus to restore. Martin Hogbin (talk) 17:05, 7 July 2010 (UTC)

But there was no consensus to remove it either. -- Rick Block (talk) 22:13, 7 July 2010 (UTC)

But Rick, you are the only person who even claims to know what it means. Martin Hogbin (talk) 09:56, 8 July 2010 (UTC)

Falk is an relevant author on the subject and therefore imho should be used for the article. Her work seems to be in particular important to provide some insight in the problems of understanding MHP with regards to conditional probabilities. However I find the current description of Falks argument somewhat misleading/confusing as well and we should come up with a description that is lees prone to be misunderstood (see remarks by coffetotheorem and me in the earlier discussion in the archive). I don't have access to Falk's paper myself to propose an alternative description. However Falk's arguments are also rehashed and interpreted in Rosenhouse's book (p. 137-138, 143-145), so we probably could use Rosenhouse as additional source and to come up with a better summary of Falk's views.--Kmhkmh (talk) 23:24, 7 July 2010 (UTC)

I have nothing against putting relevant work by Falk in this article, you will note that I left the earlier two sentences in place. This is because it is clear what confusion is being referred to and what the authors were saying about it. In the case of the removed section nothing is clear. For anything that we put in 'Sources of confusion' we must be clear on, what confusion we are addressing and whether such confusion is common and relevant to the problem; what the source says about the subject; and whether what is said is in conflict with other reliable sources. I have tried to continue the discussion started by Coffee2theorems but to no avail - see the questions that I have asked above. What do I do? Start yet another section below or ask you to reply to the questions that I have already asked above. I have never said that nothing more by Falk can ever go in the article, just that what we do put in should be clear, understandable, relevant, and correct. Does anyone disagree with that? Martin Hogbin (talk) 09:56, 8 July 2010 (UTC)

The problem is among others that there are apparently disagreements on what is "clear, understandable, relevant, and correct". Anyhow following Rosenhouse's description of Falk's (and a few others) arguments she argues essentially the following. Humans tackle problems like the MHP by applying intuitive strategies they know to be working in other probability scenarios - among them in particular:

• (1) uniform probability between all options
• (2) no new information, no changes of the probabilities

Now for the MHP (1) and (2) can appear conflicting, since (1) could be used to argue the probability is 1/2 (assigning uniform probabilities to the 2 options) whereas (2) gives you the correct 2/3. According to Falk now the fact that (1) and (2) yield different results rather than the same make the problem so unintuitive. Furthermore she argues that none of the intuitive strategies is foolproof and can easily be misapplied, therefore a careful Bayesian analysis might be needed to assure the intuitive strategy is applied correctly. For (1) we've seen that it leads to false result above and for (2) there's the following comment in Rosenhouse: "The no news argument is fine as far as it goes, except that in practical situation it is often difficult to discern, absent a thorough Bayesian analysis and a meticulous attention to detail, whether or not we have learned actually learned anything." and a little later on all the intuitive strategies including (1) and (2): "All these secondary intuitions are plausible and in many situations they lead to accurate conclusions. As general rules, however, they fall flat". This is actually somewhat similar to what coffeetotheorem and I have pointed out in the archived discussion already. So to summarize the whole thing. The unintuitive nature of the problem is due to conflicting intuitive strategies and in doubt any of those strategies may have to verified by a thorough Bayesian analysis.--Kmhkmh (talk) 10:57, 8 July 2010 (UTC)

Your last sentence is fine and, if you want to say something like that I have no real objection except that it is fairly obvious. I am talking about the specific relevance ofthe no-news argument, because this is the subject of section that I removed. Before we can discuss people's confusion surely you will agree that we must agree on what they are confused about? Martin Hogbin (talk) 11:54, 8 July 2010 (UTC)

It's not about what we might agree on or not, but what a reputable sources claims to cause confusion. I didn't comment on the removed text either, but I merely pointed out which point Falk is putting forward according to Rosenhouse, since apparently it was not clear to you or others what Falk seems to claim and many don't have access to Falk's original paper nor possibly to Rosenhouse's book. As far as the no news argument goes Falk and Rosenhouse seem to argue although being correct for the standard MHP it is not necessarily obvious. The specific relevance of the no news argument is that as an intuitive strategy it conflicts with another one. I.e. it is needed to explain why the problem is unintuitive to people rather than to explain why people compute the false solution. To explain the false computation alone you strictly speaking need only (1), but to argue from a more general perspective regarding the "unintuiveness" of MHP you need (1) and (2), as problems are perceived as unintuitive in general, if the intuitive strategies are conflicting rather than producing an identical solution.--Kmhkmh (talk) 12:21, 8 July 2010 (UTC)

I have nothing against using reliable sources but before we can include any material from a source, however reliable, on the subject of confusion we must understand what the confusion is that they are trying to explain, if for no other reason than to know where to fit the statement into the article. Just cutting and pasting comments from sources into WP is not a good idea; editors should always understand the points being made. Perhaps you could comment below on exactly what the confusion is that is being addresses by Falk (apart from the general point that intuition is not always to be relied upon).

Please don't confuse other editors with yourself.--Kmhkmh (talk) 13:00, 8 July 2010 (UTC)

I have no idea what you are talking about. Shall we just stick to the MHP and sources of confusion. Martin Hogbin (talk) 13:09, 8 July 2010 (UTC)

Kmhkmh, basically, you're trying to define why this problem is such a marvelous paradox. Of course it fails if the premises are slightly changed. That's not a source of confusion, though. Glkanter (talk) 11:24, 8 July 2010 (UTC)

Exactly. Martin Hogbin (talk) 11:54, 8 July 2010 (UTC)

Confusion about confusion

If we are going to discuss why people are confused in anything other than a general sense that intuition is not always to be trusted the we must at least agree on what people are confused about. Now, as far as I know, what people are confused about is the fact that the answer is 2/3 and not 1/2. Is this or is it not the confusion that we are trying to address in the article? Martin Hogbin (talk) 11:57, 8 July 2010 (UTC)

It is what you want the article (solely) to address. How other editors or various sources see that is an entirely different matter. It kinda looks like this is turning into another proxy war for "unconditional versus conditional" and which aspect gets explained exactly when. Good luck with that .....--Kmhkmh (talk) 12:57, 8 July 2010 (UTC)

If you are going to assume that everything I say is part of some dastardly plot to push the unconditional solution I guess there is nothing I can do to stop you but in fact I am just asking questions. Why not just answer my question? Is it the confusion over the correct numerical answer that you want to address or is there some other confusion and if so, what exactly is it? Martin Hogbin (talk) 13:07, 8 July 2010 (UTC)

The confusion is about how to compute the correct numerical answer rather than what the answer is. Why most people assume the answer is 1/2 is because they rely on an intuition rather than a careful analysis. Coming up with the 2/3 answer based on a different intuition (that results in the wrong answer for slight variants) is really no better. The basic confusion remains unless an approach that is actually sound is used. -- Rick Block (talk) 19:30, 8 July 2010 (UTC)

But the no news argument does give the correct result in the symmetric case. It is only when the host chooses a goat door non-randomly that new information is revealed. There are two sets of the confused, those who get the wrong answer, because of their (incorrect) intuition that probability is evenly distributed; and those who get the right answer by the 'wrong' method because, their no news is no change intuition is correct but they incorrectly believe that the host opening a door to reveal a goat can never give them any new information. Martin Hogbin (talk) 19:57, 8 July 2010 (UTC)c

And the equal probability intuition gives the correct result in the "host forgets" case. This doesn't mean either of these are good ways to approach a problem like this. Which is the point Falk (not me) is making. How about if you propose wording for this point? -- Rick Block (talk) 22:01, 8 July 2010 (UTC)

I will give it a go. There is also the question of where it should go. Maybe a section within 'Sources of confusion' relating to the conditional solution. Martin Hogbin (talk) 22:33, 8 July 2010 (UTC)

The solutions that it pertains to are the unconditional solutions, so putting it in a section relating to the conditional solution seems curious. -- Rick Block (talk) 14:32, 9 July 2010 (UTC)

Well, I guess it pertains to the fact that some consider the unconditional solution wrong (especially) in the case that the host has a known door preference. All I am really suggestion is that it is separated from the 'even distribution' confusion which relates to whey people get the answer of 1/2. Martin Hogbin (talk) 22:33, 9 July 2010 (UTC)

You seem to still be missing the point, which is that relying on intuition rather than a sound analysis is what messes people up. It's why a large number of people say the answer is 1/2 - but it's also why another large number of people (including many who originally thought the answer is 1/2) believe solutions that say things that are manifestly untrue. Intuition #1 makes it seem like "there are only two doors, so therefore the probability of each must be 1/2" must be true. Intuition #2 makes it seem like "since we already know one of the unselected doors is a goat, the host opening a door cannot change the probability of the player's initially selected door" must be true. Each of these are true in some circumstances, but neither is always true. And, again, this is what Falk is saying. Falk repeats much of this in her book "Understanding probability and statistics" (on page 187-189), which you can preview on Google Books if you'd like to read what she says yourself. Actually, even though I haven't read anything other than the first page, I strongly suspect this is what Bar-Hillel goes on to say here as well. -- Rick Block (talk) 23:37, 9 July 2010 (UTC)

And there it is. After all this time:

Rick Blocks says: "Why most people assume the answer is 1/2 is because they rely on an intuition rather than a careful analysis. Coming up with the 2/3 answer based on a different intuition (that results in the wrong answer for slight variants) is really no better." - Rick Block (talk) 19:30, 8 July 2010 (UTC)

Yes, it is. Because in this case, it's correct and valid. And it's not necessarily 'intuition'. It can be called 'thoughtful analysis'. Period. End of discussion. Go away. Glkanter (talk) 19:35, 8 July 2010 (UTC)

Continued from above

I am not sure what point you think I am missing. I agree that in probability problems, relying on intuition can lead you astray, and if that is the point you want to make then why not just say so in those words?

You then go one to mention to examples of how this is the case. The first explains how the (incorrect) 'even distribution' intuition leads people to incorrectly believe that the answer is 1/2. In your second example you show how a misapplication of the 'no news' intuition can result in a different error.

I have managed to obtain a copy of Falk's paper. The first thing to note is that she is a psychologist writing primarily about the similar but not identical 'three prisoners' problem. Having now read it, let me suggest this wording:

In addition to the "equal probability" intuition there is a competing intuition that can lead to incorrect answer in some cases. This is the assumption that the revealing of a goat by the host can never reveal any new information that might affect the probability that the car is behind the initially-chosen door. Although this intuition proves correct if the host chooses uniformly at random between doors hiding a goat it is not correct if the host is known [Falk makes clear this must be known on page 207] to choose non-randomly between such doors. In this latter case the specific door that the host opens does reveal information about the probability that the car is behind the door that the player originally chose and therefore it also affects the probability of winning by switching

I suggest that this put back as a separate paragraph in the same position as before. The note in square brackets is just for readers here. Martin Hogbin (talk) 12:16, 10 July 2010 (UTC)

The point you seem to be missing is that Falk is not saying that the answer is incorrect in some cases, but that this is incorrect reasoning - confusing cause and effect. As she says in her book [5] :

Truly, Monty can always open one of the two other doors to show a goat, and [emphasis in original] the probability of door No. 1 remains unchanged subsequent to observing that goat, still, it not because [emphasis in original] of the former that the latter is true. The probability of winning the car by sticking with door No. 1 remains unchanged due to a specific combination of priors and likelihoods characterizing this problem.

It is not quite clear what Falk is talking about here. Is she saying that, in the case that host chooses a random goat door the fact that Monty can always open one of the two other doors to show a goat does not mean that the probability is unchanged? If so, on what basis does she make this claim. Martin Hogbin (talk) 23:34, 10 July 2010 (UTC)

So, how about:

In addition to the "equal probability" intuition there is a competing deeply rooted intuition that is the basis for some solutions that lead to the correct answer for the standard interpretation of the problem but an incorrect answer for slight variants. The intuition is that revealing information that is already known does not affect probabilities. Although this is a true statement, it is not true that knowing the host can open one of the two unchosen doors to show a goat necessarily means that opening a specific door does not affect the probability that the car is behind the initially-chosen door. If the host chooses uniformly at random between doors hiding a goat (as is the case in the standard interpretation) this probability indeed remains unchanged, but if the host can choose non-randomly between such doors then the specific door that the host opens does reveal information about this probability. Solutions based on the assertion that the host's actions cannot affect the probability that the car is behind the initially-chosen door, although very persuasive, are therefore not mathematically sound (Falk 1992:207).

The last sentence is essentially your POV. Falk does not say this in her paper. Martin Hogbin (talk) 23:34, 10 July 2010 (UTC)

We could also reference Falk's book, but since the paper is earlier and says effectively the same thing I think the reference to the paper is marginally better. -- Rick Block (talk) 16:12, 10 July 2010 (UTC)

Where in the paper does Falk make the point you claim? Martin Hogbin (talk) 23:34, 10 July 2010 (UTC)

It is the entire topic of the section starting on page 207. The "persuasive" bit is from the second sentence: "If, on the other hand, we grade assumptions by their persuasiveness, no-news has no rival." The bit about not mathematically sound is a paraphrase of the contents of the entire section. For example, on page 208 (speaking directly about three prisoners here, but what she is saying clearly applies to the MHP as well - and if you want to argue this we could reference her book if you'd like): "In the original version of the problem [that is, the three prisoners problem], the warden can always truthfully name at least one of Tom and Harry, and [emphasis in the original] the probability of pardon for Dick does not change. Yet, as we have seen, is not because [emphasis in the original] of the former that the latter is true." This quote is nearly identical to what she directly says about MHP in her book (as quoted above). -- Rick Block (talk) 04:23, 11 July 2010 (UTC)

Now we have to parse the differences and similarities between the MHP that the article is about, and the 'similar', according to some, 'Three Prisoners Problem'? No thank you. I've heard about the similarities for 2 years now. But I haven't heard about any goats, or door #3 must be open, or game shows, or contestants SoK. Is there a 50/50 host bias when two goats are behind the doors? They don't sound 'similar' to me. And as Rick Block has demonstrated repeatedly his difficulties in understanding the point of various MHP articles, I have no interest in having to additionally read 'Three Prisoner Problem' articles in order to refute Rick's unsupported conclusions. 'Three Prisoners Problem" sources do not belong in this article. Glkanter (talk) 04:57, 11 July 2010 (UTC)

Rick, there is no doubt that if the host is known to choose uniformly at random between goat doors, no new information about whether or not the car is behind the originally-chosen door is revealed. This fact has been agreed by several editors here and denied by none.

True, but has to be proven, and hence the simple solution fails.Nijdam (talk) 11:47, 11 July 2010 (UTC)

It can be proven, without resort to conditional probability, by symmetry or information theory.

No it can't, because what is proven is that the conditional probability is equal to the unconditional (not mentioned the condition of the first choice). This has been explained to you several times. Nijdam (talk) 15:22, 11 July 2010 (UTC)

Exactly, the conditional probability is equal to the unconditional, this is proved without resort to a conditional calculation. Martin Hogbin (talk) 15:43, 11 July 2010 (UTC)

There is no such thing as conditional calculation! You just confirmed the shortcoming of the simple solution. The strange thing is, why don't you accept it?Nijdam (talk) 16:59, 11 July 2010 (UTC)

The simple solution works for the unconditional case, the conditional probability is proven by symmetry to be equal to the unconditional probability for the symmetrical case, thus the simple solutions solve the conditional problem for the symmetrical case. Martin Hogbin (talk) 20:48, 11 July 2010 (UTC)

You keep repeating this, but the solution you are referring to, with a proof based on symmetry, is not the (a) simple solution. There lies your problem. As soon as there is the mentioning of a proof that the conditional probability is equal to the unconditional, it is no more a simple solution. And it is precisely this lack of proof, in whatever form, where the error in the simple solutions lies. For example the combined doors solution does not mention such a proof. Come to your senses. Nijdam (talk) 21:08, 12 July 2010 (UTC)

We can now present the simple solutions and state that a reliable source (Falk) confirms that due to what she calls complete symmetry they do, in fact, solve the symmetrical conditional problem. What is wrong with that? Martin Hogbin (talk) 21

35, 12 July 2010 (UTC)

It is a well accepted and verifiable fact of information theory that a random choice conveys no information.

??I think this is put a little too simple.Nijdam (talk) 11:47, 11 July 2010 (UTC)

I am sure that I have put it rather crudely. Perhaps someone who knows more about the subject could put it better. It is nevertheless true. Martin Hogbin (talk) 14:53, 11 July 2010 (UTC)

Even then it is of no help in rescuing the simple solution.Nijdam (talk) 15:22, 11 July 2010 (UTC)

As we already know that a goat can be revealed, no information relating to whether the car is behind the originally-chosen door is revealed when the host opens a goat door uniformly at random. Despite the generally accepted truth of this fact, I have not attempted to add it to the article because some editors seem to object to it there are no reliable sources which make that exact point in specific relation to the host's door choice in the MHP.

So it is with your claim that, if the host chooses a goat door uniformly at random, a simple solution is not sound. There is a split of opinion between editors here on that matter but no reliable source at all makes that exact statement in relation to the MHP.

The split (amongst the serious editors) is you on one side and the rest on the other.Nijdam (talk) 11:47, 11 July 2010 (UTC)

Even Morgan do not actually say that the simple solution is wrong in the specific case of a known host random goat door choice.

Morgan does.Nijdam (talk) 11:47, 11 July 2010 (UTC)

I have just looked through the Morgan paper and cannot find anywhere in it where they say that the simple solution is false for the case p = q = 1/2. Most of the paper is about the more general case where q is unknown.

Read what they say about the solution F1. Nijdam (talk) 15:22, 11 July 2010 (UTC)

Solution F1 states 'regardless of the host's action'. I am talking about the case where the host is known to choose a goat door uniformly at random. Martin Hogbin (talk) 15:43, 11 July 2010 (UTC)

By the way, do you now agree that Morgan's solution is as incomplete as the simple solution, for the reasons given above in the section, 'One for the pedants'? I notice you kept very quiet about that subject. Martin Hogbin (talk) 14:53, 11 July 2010 (UTC)

I've been away, and just try to get known with all that's been done here. But, to your question, no. Nijdam (talk) 15:27, 11 July 2010 (UTC)

No doubt you will explain further. Martin Hogbin (talk) 15:43, 11 July 2010 (UTC)

I will in due course.Nijdam (talk) 16:59, 11 July 2010 (UTC)

Falk's paper is rather vague on her second point, she sometimes refers to the 'no-news' belief and at others she refers to the 'apparent no-news situation' My version of this point is clear and neutral, making no contentious claims not clearly supported by the sources. I do not think any editor here would say that my version is incorrect. You have added your personal opinion on the matter but this is not supported by a clear statement on the subject by Falk or other reliable source. In cases of dispute, we must only put what the source actually says. Martin Hogbin (talk) 09:37, 11 July 2010 (UTC)

Pure unsupported 'junk science' and bias above from Nijdam. He and Rick Block should both be banned from editing this article. Glkanter (talk) 11:53, 11 July 2010 (UTC)

It's obvious, as is well known. Nijdam, piping up with improper remarks, does not like to distinguish between TWO clearly differing variants:

The one without any presumption regarding the host's behavior (allowing for a "Suppose he is biased"), that's the one he's after,

and the second one, expressly describing the behavior of the host (if he has the choice): That, accordingly, the host always will open one of his two doors, both hiding goats, uniformly at random, without any "bias" (q=1/2). Not giving away any further info regarding the chance of the door selected by the guest.

Rather viewless. --Gerhardvalentin (talk) 15:38, 11 July 2010 (UTC)

In addition:

1. Even if we allow "Suppose he is biased", that does not mean the contestant is aware of this. The puzzle begins, "Suppose you are on a game show...". That mean's it's the contestant's State of Knowledge that is of interest. Hosts don't tell contestant's if they have a 'bias'.
2. Carlton's decision tree shows that any (possible or not) 'host bias' is immaterial to solving the problem.
3. It can still be argued that 'contestant's indifference' forces the 'host bias' to be equivalent to 50/50. Except that the Morgians have argued exactly the opposite all along.

These "Suppose he is biased" views have been, unfortunately, published. They go in the article. But not as an editorial POV, as has been the case for 5+ years. A 'Controversies' section is needed for Morgan, Falk, etc... Glkanter (talk) 17:00, 11 July 2010 (UTC)

Once again, directly quoting from Falk's book [6]:

Monty's problem, just like another well-known isomorphic problem, the problem of the three prisoners, brings to the fore people's intuitive beliefs. Another widespread belief, which is incompatible with the uniformity assumption, is the no-news-no-change belief: since you know from the beginning that Monty intends to open one of the doors you don't choose to reveal a goat, and since he can always do it, observing the goat behind door No. 3 provides no new information and should induce no change in your views. At face value, there is something very compelling about the idea that when we receive a piece of information we have known all along, it should not alter our assessment of the situation. This view is reinforced by the Bayesian calculation showing that when incorporating the observation of the goat behind door No. 3, the probability of the car being behind door No. 1 still says unchanged (it is also expressed by Marilyn in Parade Magazine, December 2, 1990, p. 25, and by Gardner, 1992). It is nevertheless at fault [emphasis added].

This is the paragraph immediately before the paragraph I've already quoted above from the same source. It is quite clear what Falk is saying here, and it is not that the answer is incorrect for some formulations of the problem but that the reasoning is incorrect. -- Rick Block (talk) 16:11, 11 July 2010 (UTC)