Talk:Monty Hall problem

Totally overthought this problem

There's no need for all these mathematical proofs. If you stick, you have a 1/3 chance of picking the prize. If you switch, however, you are trying NOT to pick the prize. By switching when you do not win the prize, you will win every time. This means a 2/3 chance of winning. — Preceding unsigned comment added by 94.7.196.157 (talk) 01:23, 21 February 2012 (UTC)

You are quite right, of course, and you are not the first person to point this out. Martin Hogbin (talk) 09:38, 21 February 2012 (UTC)

Yep, it really is very simple. Yet the majority of people get it wrong when they first encounter it. That is what makes a good riddle: the answer is only obvious when you see it. And this is what makes the literature, and this article, so overwrought: explaining the obvious is no simple matter. ~ Ningauble (talk) 20:14, 21 February 2012 (UTC)

Risen from death

My ban has expired, and I wonder (wondered for the last year) that the simple solution Devlin gave (the combined doors reasoning), is still present. Even knowing Devlin himself has come to the insight it's flawed. Nijdam (talk) 22:04, 27 March 2012 (UTC)

In what way is it flawed? We can discuss this on my talk page if you wish. Martin Hogbin (talk) 22:26, 27 March 2012 (UTC)

Maybe it suffices to mention that, as you know, every door has the same probability 1/3 on the car (in the standard version). Hence it is impossible to reach any conclusion that says different. Nijdam (talk) 22:36, 28 March 2012 (UTC)

The flaw is that there is no law in probability that lets you group two doors together such that their combined probability before and after the host opens one of them remains constant. Assuming the car is located at the outset randomly (uniformly distributed), then p(door 1) = p(door 2) = p(door 3) = 1/3. Any two of these sum to 2/3. After the host opens a door there are now a new set of conditional probabilities. If the host opens door 3 the new probabilities are p(door 1|host opens door 3) = ?, p(door 2|host opens door 3) = ?, and p(door 3|host opens door 3) = ?. We indeed know p(door 3|host opens door 3) is 0, but nothing we've said (so far) says anything about the sum of this and p(door 2|host opens door 3). p(door 2) + p(door 3) is 2/3, but this has nothing do to with p(door 2|host opens door3) + p(door 3|host opens door 3).

Examining Devlin's quote in detail (Devlin calls the doors A,B, and C with A being the one the player picks and C being the one the host opens):

1) There are two doors you did not choose, and the probability that the prize is behind one of them is 2/3.

Right. p(door B) + p(door C) = 2/3

2) I'll help you by using my knowledge of where the prize is to open one of those two doors to show you that it does not hide the prize. You can now take advantage of this additional information. Your choice of door A has a chance of 1 in 3 of being the winner.

Right. p(door A) = 1/3, in the same way that p(door B) = p(door C) = 1/3

3) I have not changed that.

Well, actually, you have. You've changed p(door A) into p(door A|host opens door C). The value of this conditional probability is the whole point of the problem. If what is meant here is that p(door A) is the same as p(door A|host opens door C) some sort of argument is needed since this is neither obvious nor even necessarily true given what has been said so far. My assumption is most people when they read this (perhaps even Devlin when he wrote it) are thinking it is talking about p(door A), i.e. door A's probability of hiding the car before the host opens any door. This is indeed still 1/3, but p(door B) and p(door C) are also still 1/3. If we want to talk about probabilities after the host opens door C, we must be talking about one of two sets of conditional probabilities. One set conditioned on the host opening door B (which could have, but didn't, happen from our initial starting point), the other conditioned on the host opening door C. By opening door C the host has changed each door's probability into a conditional probability (conditioned on the host opening door C).

So, the probability that door A hides the car at the start is 1/3. We all agree about that.

What value does it change to after the host has opened door C?

Or, if you prefer: P(Car=A)=1/3. What is P(Car=A|host opens door C|the host chooses uniformly between legal doors)? 1/3? More than 1/3? Less than 1/3? Martin Hogbin (talk) 18:07, 5 April 2012 (UTC)

Where, pray tell, did "|host chooses uniformly between legal doors" come from? You seem to be saying there is no distinction between P(Car=A) and P(Car=A|host opens door C) if they happen to have the same numeric value. Or perhaps you're saying it doesn't matter if the logic is confused as long as it ends up with the correct numeric result. Again, the whole question is the relationship between P(Car=A) and P(Car=A|host opens door C), or (equivalently) between P(Car=B) and P(Car=B|host opens door C). Per Falk, there are two very strong, competing, intuitions:

1) The host cannot change the initial probability the car is behind the door the player picks, i.e. P(Car=A) must have the same value as P(Car=A|host opens door C), which implies P(Car=B) must be different than P(Car=B|host opens door C), or

2) Since the car was equally likely to be behind each door at the beginning, with only 2 doors left it must be equally likely to be behind the remaining doors, i.e. P(Car=A) = P(Car=B) implies P(Car=A|host opens door C) = P(Car=B|host opens door C)

IMO, glossing over the distinction between P(Car=A) and P(Car=A|host opens door C) is glossing over the entire problem. Most people's initial intuition is #2. A different intuition leads to #1. Relying on either of these results in the wrong answer depending on the exact problem set up. -- Rick Block (talk) 19:55, 5 April 2012 (UTC)

"host chooses uniformly between legal doors" is part of the standard setup.

Perhaps now you could answer my question is the new (conditional) probability 1/3, >1/3, or < 1/3? Martin Hogbin (talk) 20:03, 5 April 2012 (UTC)

Part of the standard setup according to whom? The answer to your question is (of course) 1/3, but this doesn't make P(Car=A) the same as P(Car=A|host opens door C) any more than 2 apples is the same as 2 oranges. I am really not in the least interested in arguing with you about this (we're long past the point where any discussion between us is likely to be productive). Although I agree they have the same numeric value, you are clearly not going to convince me P(Car=A) is mathematically the same as P(Car=A|host opens door C) and I am clearly not going to convince you that they're mathematically different. -- Rick Block (talk) 06:37, 6 April 2012 (UTC)

The host chooses a legal door uniformly according according to K&W, the article itself, Nijdam and probably you. It is universally accepted that in the standard version of the problem the host is taken to choose uniformly between legal doors. If you like, I can spend a few paragraphs on the arguments page going through old ground again to make this clear but I suspect you are just trying to avoid the obvious conclusion shown below.

So you agree that the probability that door A hides the car given that the host has opened door C is 1/3. You will note that this value has (obviously) not changed from its initial value, thus Devlin's argument is fine.

I fully understand that P(car=A) is conceptually different from P(car=A|host opens door C) which is conceptually different from P(car=A|host opens door C|host chooses uniformly between legal doors) which is conceptually different from the actual conditional probability that we wish to calculate, which is P(car=A|host opens door C|host chooses uniformly between legal doors|host says the word 'pick'|loads of other stuff that clearly does not affect the numerical value of the probability that we wish to calculate). Martin Hogbin (talk) 08:42, 6 April 2012 (UTC)

So you agree that P(car=A) is conceptually different from P(car=A|host opens door C), but so long as they have the same numeric value (which is forced by additional constraints on the host in the explicit, "standard" version often omitted in popular versions) you see this difference as being no more important than conditioning on the specific words the host uses and have no problem with arguments that mathematically replace one with the other (if we call these X1 and X2, the "combining doors" argument is effectively X1=1/3, X2+Y2+0=1, so [???] 1/3+Y2=1, so Y2 must be 2/3 QED)? I have a question I'd like you to ponder (don't feel compelled to respond here) - what exactly is the critical difference between the situations where everyone completely understands the car is behind door 1 or door 2 with probability 1/3 (i.e. before the player makes her initial pick, and after the player makes her initial pick but before the host opens anything) and the situation nearly all people have trouble with? To be more specific, why is it obvious that P(car=A) = P(car=B) = P(car=A|player picks A) = P(car=B|player picks A) = 1/3, but not obvious what the value is of P(car=A|player picks A|host opens C) or P(car=B|player picks A|host opens C)? Hint: it has something to do with the sample space. -- Rick Block (talk) 16:34, 6 April 2012 (UTC)

The fact is that when Devlin says that the host opening door C does not affect the probability (obviously meaning its numerical value) that the car is behind A he is correct. I find that obvious, you may not, but it is in a reliable source and it is factually correct. There is therefore no reason to remove it from the article.

4) But by eliminating door C, I have shown you that the probability that door B hides the prize is 2 in 3.

No. By opening door C he's shown you the conditional probability of door C is 0. You can't take this 0 and substitute it for p(door C) in statement #1 since p(door C) and p(door C|host opens door C) are completely different things (apples and oranges).

We've been over this so many times I find it hard to believe anyone does not understand it. -- Rick Block (talk) 15:01, 5 April 2012 (UTC)

• ^{[citation needed]} Where does Devlin himself report that the insight of combining the un-chosen doors is flawed, and in what respect does he say that it is flawed? ~ Ningauble (talk) 18:31, 5 April 2012 (UTC)

See http://www.maa.org/devlin/devlin_12_05.html. After some somewhat still confused attempts at providing an "intuitive" explanation (distinguishing the standard problem and what amounts to the "host forgets" variant), he says "Confused? As sometimes arises in mathematics, when you find yourself in a confusing situation, it may be easier to find the relevant mathematical formula and simply plug in the appropriate values without worrying what it all means." He then proceeds to show how to find the probabilities in both variants using Bayes' formula. -- Rick Block (talk) 19:55, 5 April 2012 (UTC)

Not quite retracting his simple explanation is it? More like saying a more complicated solution is required for a more complicated problem. Martin Hogbin (talk) 20:05, 5 April 2012 (UTC)

He's contrasting two simple versions of the problem, where the "intuitive" reasoning that produces the "correct" answer for the standard version fails for the other version (for reasons that are not obvious). Is he retracting his simple explanation? No. But he's definitely admitting it's flawed ("confusing") and that an easier approach to the correct answer may be to use Bayes' formula. -- Rick Block (talk) 06:37, 6 April 2012 (UTC)

We could argue forever about what Devlin was thinking but, whatever you might be taking him to mean, it is not a clear enough retraction of his explanation to justify the removal of a section closely based on a reliable source. Martin Hogbin (talk) 08:26, 6 April 2012 (UTC)

Ironically, a plug-and-grind approach "without worrying what it all means" may be persuasive, but it neither explains nor justifies anything. Anything becomes less "confusing" if one accepts an invitation to take it on faith. The application of Bayes' formula (or other formalisms) is correct, and the combined doors approach is correct; but in both cases the underlying concepts, and the method of their correct application, elude most people. Hence, I think Devlin is very apt in his closing remark: "So there you have it. Whether you believe it is another matter." ~ Ningauble (talk) 16:03, 6 April 2012 (UTC)

"Correct" is actually not the issue - combining doors is a published solution. The question is whether we should editorially choose to include it or not, and (if so) whether we should also include various published caveats or criticisms and (again, if so) whether these caveats or criticisms should be presented essentially inline. Call it what you will, Devlin says his intuitive solution appears to apply to the commonly posed "host forgets" variant but is not valid in this case. It's perhaps interesting to note that the difference between the "host forgets" (or "host doesn't know where the car is") version and the so-called "standard" version was the original point of Whitaker's letter to vos Savant. -- Rick Block (talk) 16:34, 6 April 2012 (UTC)

That he suggests combining doors may be confusing, in that it does not answer a different problem, is not a substantive caveat or criticism. That he chose not to remove potential confusion by explaining how to apply the method correctly is regrettable. A little explanation of what it means might have been in order, but his agenda was to present a different method. The point of my post above was that his non-explanation is not only regrettable, it is risible when he asks us ignore the meaning of the method he presents. His exercise in meaninglessness is not a serious caveat or criticism. ~ Ningauble (talk) 19:53, 10 April 2012 (UTC)

Is this[1] the source article you are all talking about? If so, I don't see anything here that could be considered a "retraction". I interpret it as Devlin simply using the problem's alternate version of a third party randomly chosing another door to illustrate the key to understanding the problem; namely, the fact that Monty knows which door the car is behind and always opens a door that reveals a goat in the standard version. Devlin goes on to explain in detail how the alternate version results in the naively intuitive reasoning of "makes no difference if I switch" is actually correct...for the alt version only. There is nothing earth-shattering, surprising or confusing about this to me. It is crucial to completely understanding the original problem in the first place. On a related note regarding the orginal problem, I don't think Devlin did a very good job intuitively explaining the original problem. In my opinion, the best way is to point out to the reader that by switching, the only way you can lose is if you happen to select the door with the car behind it initially. Probability 1/3 of course. --RacerX¹¹ Talk to me^{Stalk me} 17:06, 6 April 2012 (UTC)

Exactly, although I would point out that some people do find Devlin's solution intuitive and convincing. Martin Hogbin (talk) 20:55, 6 April 2012 (UTC)

I even will go further interpreting Devlin's words. Devlin and many others are fully mistaken in their simple combined doors idea. And it is my opinion Devlin has come to this insight himself too, but is not willing to plainly admit this, so gives some misty formulation to cover up. Nijdam (talk) 08:12, 7 April 2012 (UTC)

Nijdam, that is an interesting point you make about people not wanting to admit their mistakes, and yes it does happen. I would say that the Morgan paper is a good example. My guess is that at least one author of the paper got the answer wrong (in fact 1/2) and, rather than admit their mistake, they decided to concoct a formulation is which 1/2 could be a correct answer. This ruined a simple and unintuitive problem that most people get wrong. To their credit, they have almost admitted this in response to our letter. Martin Hogbin (talk) 08:56, 7 April 2012 (UTC)

Can't follow you here. Nijdam (talk) 06:15, 8 April 2012 (UTC)

Let me explain, I fully admit that this is idle speculation on my part. Most people, when asked for the probability that a player who swaps will win the car, will give the incorrect answer of 1/2. My suggestion is that at least one of the Morgan authors gave this wrong answer to one of their colleagues. Having had their mistake pointed out to them and not wishing to admit that they were wrong, they then tried to come up with a plausible formulation in which the correct answer was indeed 1/2. This, of course, is the contrived case where the player chooses door 1 and the host always opens door 3 when permitted by the rules. This contrived host bias forms the basis of their paper. In their response to our letter they agree that 'the answer' was, in fact 2/3. Martin Hogbin (talk) 09:30, 8 April 2012 (UTC)

I can't say it isn't true, but I very much doubt it, as in that case this possibility would be emphasized more. And from what you wrote before, I got the impression you thought the article was meant to criticize vos Savant and the simple solution, which you didn't like. But anyway, one is entitled to change their opinion. As you'll know I value the article as it makes clear the distinction between the "unconditional" interpretation of the problem and the standard form, and the respective solutions. Nijdam (talk) 18:40, 8 April 2012 (UTC)

I would add that the common emphasis on a specific example case ("You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, 'Do you want to pick door No. 2?'), and the fact that this changes the sample space in an unexpected way (in the reduced sample space door 1's conditional probability retains the same numeric value as its unconditional probability while door 2's conditional probability doubles) is an integral part of this problem. Of course, it can be posed in an unconditional kind of way - for example by suppressing the knowledge of which door the host opens (which is extremely artificial given the gameshow context), or changing the question from what would you do in a specific case to what is the best strategy to pre-select before going on the show - but this is not in the least typical, and (per Krauss&Wang) is not how nearly everyone interprets the common problem statement. Although they certainly could have presented it in a less hostile way (as others have, for example Gillman, and Grinstead&Snell, and Rosenthal, etc.), I think Morgan et al.'s criticism of vos Savant (for using an unconditional solution for what at least seems to be intended to be, and most people interpret as, a conditional problem) is spot on. Regarding their supposed "retraction" - please read Coffer2theorems reply about this in the archives. I completely agree.-- Rick Block (talk) 06:06, 9 April 2012 (UTC)

Sockpuppet Investigation

2 more IPs blocked.[2] Drop me a line on my talk page if any more IP editors show up who just happen to have strong and biased views on a year-old arbcom case. --Guy Macon (talk) 22:26, 7 April 2012 (UTC)

Guy, is there any reason that Glkanter should not be allowed to return to editing WP now that the other's bans have expired? Martin Hogbin (talk) 16:44, 10 April 2012 (UTC)

If you look at Wikipedia:Arbitration/Requests/Case/Monty Hall problem#Remedies you will notice that most remedies are for a period of one year but one - a topic ban on MHP for Glkanter - is indefinite. In addition, If you look at User:Glkanter you will see that he is indefinitely blocked from all editing (post-arbcom admin decision, not arbcom ruling). He is, of course, free to request that the block be removed, but to do that he will have to convince an uninvolved admin that he will comply with Wikipedia's policies. --Guy Macon (talk) 22:51, 10 April 2012 (UTC)

Suggestion

Rather than argue about this ad infinitum, is there something productive we might be able to do? For example, how about changing the "Simple solution" section to a section that explicitly addresses what many authors present as a simpler problem, i.e. whether you should use a preselected switch or stay strategy changing the decision point from after the host opens a door to the very beginning of the game (much in the same way Grinstead&Snell treat it, and the same way Carlton suggests)? I think this has the advantage of progressing from simple to more complex (which, as I understand it, is Martin's goal), but it also alerts the reader that what follows may not totally address the problem they're currently thinking about (easing the shift to thinking about the "unconditional" problem). This is, more or less, the same structure the article had following its last successful FAR - but rather than (as the article did then) transition from a simple to more complex solution by saying the simple solution doesn't quite address the problem as stated, presents an explicitly different problem that stone-cold simple solutions definitely address (the choice is always stay or always switch, if you always stay you win 1/3 of the time so if you always switch [to whichever door the host opens] you must win 2/3 of the time). Maybe we'll argue about which simple solutions should be included in this section, but I think this could make the article much more to the point, significantly shorter, easier to understand, and far less contentious. -- Rick Block (talk) 06:06, 9 April 2012 (UTC)

Martin's comments

At first sight that might look like a new and positive suggestion, which I would welcome, but it is, in fact, just a restatement of your original position, that we should start with the simple solution provided they contain a 'health warning' that they do not answer the question as asked.

The only real consensus we have ever had here to start with the simple solutions, as given in reliable sources, and then to state the objections to those solutions that some sources give, and then to give the Morgan-style solutions given by some other sources. This does not push any POV but simply does what all good text books and encyclopedias do which is to start simple, then mention any complications, then discuss the more complicated aspects.

If you like, I can show you where there was a consensus to do this. Unfortunately you and a few other editors have consistently ignored this in favour of your own POVs. Martin Hogbin (talk) 08:28, 9 April 2012 (UTC)

Me however, I fully support Rick's suggestion. And I do not like to be accused of pushing forward my own POV, as such POV is found in reliable sources. Nijdam (talk) 09:30, 9 April 2012 (UTC)

Yes, of course, but some reliable sources say different without imposing the view that Morgan is correct and other sources are wrong. Martin Hogbin (talk) 10:04, 9 April 2012 (UTC)

Your claim about consensus is at best dubious (as I recall, it was after nearly all the editors who favor featuring a conditional approach had stopped commenting here). My suggestion does not just seem different, it actually is different. The idea is fundamentally to present material that no source disputes. No source disputes the simple 1/3 if you stay, 2/3 if you switch solution if you move the decision point to before the host opens a door (in fact, several sources such as Gillman that have issues with the simple solutions suggest this variation). No source disputes that the conditional probability approach to the standard version is correct. We could even add a transition section (with content no source disputes) explaining that the "simplified" problem must have the same answer as the non-simplified problem as long as the problem is symmetrical (which is forced by the constraint on the host that he open an unchosen door randomly [uniform] if the player's pick happens to hide the car). The structure would be:

• Lead (more or less as is)
• Problem description (more or less as is)
• Related, simpler problem (new section, presenting the version and its stone cold simple solution where the decision is between the strategy of staying vs. the strategy of switching - referencing Grinstead&Snell and/or Carlton and/or Gillman and/or Gill etc.)
• Solution
  • New section (explaining why the standard problem where the decision is after the host opens a door must have the same answer as the simpler problem if the problem is symmetrical - referencing Falk and/or Gill)
    • Other simple solutions based on the premise that the problem is symmetrical (if you want - although I suspect this is in all likelihood not even necessary)
  • Solution using conditional probability (another approach, showing the same result, presented in nearly all introductory probability textbooks, is ...)
    • Informal
    • Formal
  • Solution using game theory (yet another approach, where the problem is viewed as a 2-player game ...)
• Variants
• History

No health warnings. No POV pushing (in either direction). -- Rick Block (talk) 20:10, 9 April 2012 (UTC)

Essentially you want to say, right at the start that the simple solutions do not solve the MHP in its normal form. Most sources say that it does. Martin Hogbin (talk) 21:10, 9 April 2012 (UTC)

No. I want to say right at the start that there is a related problem that inarguably permits a very simple solution - followed immediately with why it is that this solution also solves the MHP in its normal form (assuming symmetry). Rather than this be a discussion between us, I'd be interested in comments from others as well. Anyone else have any opinions about this? -- Rick Block (talk) 01:13, 10 April 2012 (UTC)

I am not quite sure what you are suggesting then. The most well known statement of the problem is that by vosSavant/Whitaker. If you are suggesting that we offer the simple solutions as solutions to this problem that is fine by me. If, on the hand you are suggesting that we propose some different problem to which the simple solutions are an answer that this is just our original position restated.

Most sources offer simple solutions to the vosSavant/Whitaker formulation. Martin Hogbin (talk) 08:34, 10 April 2012 (UTC)

Is the outline above not clear? I'm suggesting adding a NEW section between the "Problem" description (where the standard problem is described) and the "Solution" section, and in this new section describing the related, simpler problem (posed and answered by several sources) where the question is changed to "what is the best pre-selected strategy to follow, always stay or always switch" which explicitly moves the decision point to before the host opens a door. That the answer to this question is 1/3 chance of winning the car for the staying strategy, and 2/3 chance of initially selecting a goat (and, thus, winning the car) for the switching strategy seems to be intuitively obvious to most people. I'm suggesting including in this section only this one, extremely intuitive, solution (the one provided by sources that pose this alternate problem - not saying anything in this section about any sources that present this same solution or any other solution to the standard problem). Then, I'm suggesting starting the immediately following "Solution" section (which addresses the standard problem), with a section explaining why the answer to the standard problem must be the same as the answer to the simpler problem.

The point of this is to lead the reader to the insight that the intuitive 1/3:2/3 answer of the "always stay/always switch" argument actually applies to (and is correct for) the standard problem. Any other solutions we include would follow this new material.

If you'd like, I could write a draft. And, again, I'm still interested in other opinions about this. -- Rick Block (talk) 15:46, 10 April 2012 (UTC)

Rick, you seem to think that saying exactly the same thing in different words makes it a new idea. You want to have a different question (that is to say not the standard MHP) that the simple solutions do answer. That is exactly the same as saying that the simple solutions do not answer the standard MHP. Most sources do not support this view.

There is a compromise which was accepted as a consensus, and that is to have the simple solutions first, without health warnings or qualifications or alternative questions, and then to have a description of why some sources consider these solutions to be incomplete, followed by the Morgan-style solutions. This is how the rest of the world does things. Martin Hogbin (talk) 16:53, 10 April 2012 (UTC)

Would you please read what I'm actually saying? You apparently either don't understand or are completely misinterpreting. -- Rick Block (talk) 19:04, 10 April 2012 (UTC)

You seem to make it fairly clear above. See my comments in italic below. Have I got it right? Martin Hogbin (talk) 22:22, 10 April 2012 (UTC)

No. I've marked up your text below (and see the explanation further below as well). Is this more clear? -- Rick Block (talk) 06:35, 11 April 2012 (UTC)

• Lead (more or less as is) <-- yes.
• Problem description (more or less as is) - That is the standard vos Savant/Whitaker statement <-- not quite. It's the K&W explicit version (the "standard" version, per sources like Barbeau) including constraints on the host omitted from the vos Savant/Whitaker version, in particular including the constraint that the host must choose randomly between two "goat doors".
• Related, simpler problem (new section, presenting the version and its stone cold simple solution where the decision is between the strategy of staying vs. the strategy of switching - referencing Grinstead&Snell and/or Carlton and/or Gillman and/or Gill etc.)

So that is a different problem? Not the standard one <-- Yes. But it is not one that anyone is going to say "simple solutions" generically solve. It's one for which the sources that present it include a solution. That solution, and only that solution, will be presented in this new section. Perhaps you're confused by the fact that this solution is also presented (by other sources) as a solution to the standard problem.

• Solution To the the non-standard problem. <-- no. The new section above would include the one and only solution provided by the sources presenting the non-standard (simpler) problem. This section returns from the slight diversion (I'm imagining the new section is about two paragraphs - considerably less text than this section of the talk page - it would actually be easier to draft it than to keep discussing it like this) to the "standard" problem and presents its solutions.

So, Martin, do you understand now? Although you might not think so, I do want your comments as well. -- Rick Block (talk) 14:52, 11 April 2012 (UTC)

The related simpler problem is an insignificant and un-notable problem that is of no interest to anyone except as an introduction to the Morgan-style solutions. It will be utterly confusing to most readers to have the problem split into two before it has be solved. Martin Hogbin (talk) 19:28, 11 April 2012 (UTC)

Any other opinions about this suggestion?

I would really like comments about this idea from folks other than Martin (and I would appreciate it if Martin refrained from commenting in this section). -- Rick Block (talk) 19:04, 10 April 2012 (UTC)

(moved from just above)

I'd like to see the reliable sources supporting the prominence in the article for such an approach. Do you believe this makes the article more readable to the Wikipedia reader? I don't. 76.190.228.162 (talk) 15:52, 10 April 2012 (UTC)

The reliable sources supporting this approach (specifically, introducing and solving a related, simpler problem) include Grinstead&Snell, and Carlton (detailed references in the article). And, yes, I think this would make the article more readable to the Wikipedia reader. -- Rick Block (talk) 19:04, 10 April 2012 (UTC)

Hold on a minute Rick! Is this a section where I am asked to refrain from commenting but you can comment freely? Martin Hogbin (talk) 22:29, 10 April 2012 (UTC)

Exactly. I ask this because our discussions tend to get so voluminous that nobody else can get a word in edgewise. It's not that I don't want anyone to hear what you have to say, but if you could say it elsewhere my hope is others will be more likely to add their comments here. I've added a header for the section with your comments (just above). Fair enough? -- Rick Block (talk) 06:35, 11 April 2012 (UTC)

That seems rather an odd system. I can understand both of us keeping quiet for a while to let others have their say but not a section where everybody can edit except me. However I am willing to try anything for an easy life so I will keep out of this section. Martin Hogbin (talk) 14:33, 11 April 2012 (UTC)

That would the "Carlton" who is reliably sourced with a simple solution in the current article? As well as the recently "recanted" Devlin, who is also reliably sourced with a simple solution in the current article? Seems counter-intuitive, no? Especially with three active editors already disagreeing. What is your Grinstead and Snell support sourcing? Is your approach consistant with Selvin's letters? 208.54.80.173 (talk) 19:21, 10 April 2012 (UTC)

Without question, this proposal is contrary to vos Savant's writings. Would you describe this as the "predminant" theory among the vast published reliable sources? 208.54.80.173 (talk) 19:32, 10 April 2012 (UTC)

^{[citation needed]}No recantation has been entered into evidence. ~ Ningauble (talk) 23:28, 11 April 2012 (UTC)

My similar suggestion last July at Would it help to disambiguate "the" question?, to associate simple solutions with a simple interpretation of the problem, foundered on concerns about "health warnings" and on the problem of how to identify the "simple" interpretations and solutions in a neutral way. Unfortunately, the distinction I tried to use proved to be too abstract. Rick's suggestion of using a distinction between choosing a strategy and choosing a door is also an abstraction and, although it admits of a simple solution, it is an abstraction that complicates the question rather than simplifying it.

I think Martin is entirely correct that characterizing the simple interpretation as a "related problem" distinct from some imagined "standard problem" is deprecatory, all the more when the simple approach is arbitrarily defined so as to be insufficient after the door is open. As Seymann says ("Comment on Let's make a deal: The player's dilemma," American Statistician 45: 287-288), "Without a clear understanding of the precise intent of the questioner, there can be no single correct solution to any problem." The most notable distinguishing characteristic of simple interpretations of the problem is the host's indifference in choosing a goat to reveal (i.e., symmetry) or, as Seymann puts it (ibid.), "the host is to be viewed as nothing more than an agent of chance." This is a basic, simple interpretation of the problem, not an extra stipulation for the purpose of making deficient solutions contingently viable. ~ Ningauble (talk) 19:58, 10 April 2012 (UTC)

For the nth time, I'm not suggesting associating any simple solutions with the simple problem except the one (consistent) solution the authors presenting the simple problem themselves provide. What I'm precisely talking about is the section in Grinstead and Snell [3] on page 137, which poses the simpler problem I'm talking about: "We begin by describing a simpler, related question. We say that a contestant is using the “stay” strategy if he picks a door, and, if offered a chance to switch to another door, declines to do so. ..."

Their solution (the next paragraph) is this: "Using the “stay” strategy, a contestant will win the car with probability 1/3, since 1/3 of the time the door he picks will have the car behind it. On the other hand, if a contestant plays the “switch” strategy, then he will win whenever the door he originally picked does not have the car behind it, which happens 2/3 of the time. "

Simiarly, Carlton (whose "intuitive explanation" is mis-characterized in the current article as a simple solution to the standard problem) says: "Before presenting a formal solution to the Monty Hall Problem to my students, I find that it helps to give an intuitive explanation for the 1/3 - 2/3 solution. Imagine you plan to play Let's Make a Deal and employ the 'switching strategy.' As long as you initially pick a goat prize, you can't lose: Monty Hall must reveal the location of the other goat, and you switch to the remaining door - the car. In fact, the only way you can lose is if you guessed the car's location correctly in the first place and then switched away. Hence, whether the strategy works just depends on whether you initially picked a goat (2 chances out of 3) or the car (1 chance out of 3). "

Note that this is exactly the same simpler problem presented by Grinstead & Snell, with exactly the same solution.

The suggestion is to include a NEW section, including this specific simpler problem with the specific solution presented by those sources that present the simpler problem (not all "simple solutions", just the ones offered by the sources presenting this problem).

Doing this says nothing whatsoever about any other solution. It's sourced to two impeccably reliable sources (a probability textbook and a peer reviewed academic paper written by a professor of probability).

The point of this NEW section is to begin to convince the reader that switching (in the standard version, too) doubles your chances of getting the car (it's fairly obvious in this version of the problem).

AFTER this section, we'll present solution(s) to the standard problem. I'm further suggesting we start the next section, which does address the standard problem, with an explanation of why the solution to the standard problem must be the same as the solution to the simpler problem (i.e. symmetry). And only then, AFTER presenting and explaining the simpler problem, and explaining why the answer to the standard version must be the same as the answer to the simpler problem, proceed with other solutions (including simple solutions) to the standard problem.

Is there anybody here at all who understands what I'm talking about? -- Rick Block (talk) 00:14, 11 April 2012 (UTC)

Of course, I am. And I'm getting a little tired of well-meaning layman not understanding the problem. Also I like to state again that not any source on the problem is automatically reliable. For instance Devlin should not considered to be a reliable source. Richard Gill, an expert in the field, along with other sources, at least states that the simple solution is not to be considered a solution to the standard formulation, but to the simplified form. If it were up to me I'd make this clear from the start. But this is Wikipedia and there it is possible to just misinform people. Nijdam (talk) 09:35, 11 April 2012 (UTC)

The bald assertion that I do not understand the problem is unwelcome, but I would genuinely welcome an explanation of where I said something here that is factually or logically incorrect. It would also be helpful if you identified which you mean by "the simple solution," "the standard formulation," and "the simplified form." Since you appear to respect Richard Gill's expertise in this area, I will frame the remarks that follow in terms of his writings.

It is my opinion that the stipulations in Gill's Proposition 3 constitute the best choice of "simplified form" to use for introducing the subject to laypersons. Rick is essentially talking about using Gill's Proposition 1 as the "simplified form", but it presupposes understanding what unconditional really means and sparks strenuous objections that it is rather far removed from the usual narrative in which the question is posed after the door is open. I appreciate the sense in which Proposition 1 is mathematically "simple" in that it involves fewer constraints, but I am looking at it from the perspective that the constraints impose a simplifying symmetry.

I confess that my opinion in that regard is unsubstantiated, as research into laypeople's understanding of the problem has largely focused on why they jump to the incorrect 50/50 answer rather than which maths they understand. I also acknowledge that in corresponding with Richard Gill at Wikipedia, he expressed concern that the symmetry imposed by the essential constraint that distinguishes his Proposition 3 from propositions 1 and 2 may be fuzzy for laypersons;[4][5] however, he also remarked that solutions exploiting the symmetry are so obvious that it is not worth writing a research paper about it.[6][7][8] (My opinion: It is a trivial exercise in applied combinatorics for undergraduates.)

Richard did agree with me (in the first exchange linked above) that the essential difference between the main interpretations of MHP is whether or not the host is impartial with respect to which goat is revealed. The difference between these problem versions is what gives rise to endless confusion and interminable disputation. I really believe this essential difference between problem versions is best understood by laypersons in terms of the concrete stipulation of Proposition 3 about whether the host is an impartial agent of chance, rather than the much more abstract distinction about a fixed strategy of always switching as expressed in Proposition 1. In a textbook treatment of conditional probability it makes sense to frame the issue in terms of unconditional or conditional probability, before or after the door is opened; but for a layperson trying to understand what they are up against, the distinction between an impartial host and a biased one is much more fundamental and concrete. I strongly believe that the latter is the better way to introduce the difference between versions for a general audience. ~ Ningauble (talk) 23:16, 11 April 2012 (UTC)

I agree I'm suggesting introducing what amounts to Richard's proposition 1 (I assume you're talking about propositions 1,2 and 3 from [9]) - however, the way Grinstead&Snell and Carlton both phrase this avoids the unfamiliar "unconditional/conditional" jargon. Note that I'm not attempting to distinguish a "biased" host version from an "unbiased host" version (which I think makes sense only if you're already talking about an "after" situation). I really don't think laypeople have trouble with before and after (which are, of course, one of the main uses of the unconditional/conditional formalism). In particular, there isn't anyone who fails to understand that the probability the car is behind each of the doors before the host opens one is 1/3. And (IMO) the only reason the simple "1/3 if you stay vs. 2/3 if you switch" argument fails to sway anyone is that it is typically posed as a solution to the "standard" version of the problem which, with or without an unbiased host constraint, creates a context where the host has already opened a door (and, since there are only two doors left and the location of the car is still unknown, most people assume it must be a 50/50 choice between the two remaining doors). I think putting this same argument in the before context makes it accessible to nearly anyone (although anecdotal, Carlton says "it helps"). It would be lovely if there were more research about this, but I don't recollect any.

I think you're effectively suggesting distinguishing Richard's propositions 2 and 3, which I strongly suspect is fairly far beyond the grasp of most laypeople. Krauss&Wang talk about this where they say that most people assume the "standard" rules (embodied by proposition 3) whether they're explicitly mentioned or not. Even stronger, Mueser&Granberg ([10], referenced in the article) report no statistical difference between how many subjects switch given 3 substantially different problem descriptions (including one they call "standard" which is the same as what we're calling "standard" here but without the symmetry constraint, and another that is essentially the same as the "host forgets" version where the probability of winning actually is 50/50) - the implication being that even if these sorts of host constraints are provided most people simply ignore them.

The sequence I believe makes most sense (would be most accessible to a general audience) is

1) get them to understand the switch/stay probabilities are 1/3:2/3 in the "simplified" problem, which makes it clear we're talking about the before context

2) get them to understand the switch/stay probabilities in the standard (symmetrical) version must be the same as the probabilities in the simplified version (I suspect this is harder than the first step, but not nearly as hard as doing this without first establishing the 1/3:2/3 solution to the simplified problem)

3) whatever else we want to present (most people will probably lose interest)

Can you explain why you strongly think distinguishing biased vs. unbiased host would be better? You do, of course, know that host bias does not change the "always stay" vs. "always switch" probabilities (which remain 1/3:2/3 regardless of any host bias). -- Rick Block (talk) 03:38, 12 April 2012 (UTC)

It really does not matter who is right. Let us write the article for the benefit of our readers.

Let us forget which solutions are the 'right' ones. Some people think the simple ones are correct, some think the Morgan ones are correct. Both can be found in sources.

The MHP is one of the few simple problems that fools most of the people most of the time and many people do not get it even when it is carefully explained to them. We therefore need to make every effort to explain the problem and solutions as simply as possible before our readers give up. Even if I were an avid Morgan supporter I would still want to start by grabbing and holding our readers' attention by explaining the basic paradox and solution before going into the Morgan details. Here are two ways to do that:

The wrong way

• This is the standard problem (vS/Whitaker + usual (K&W) assumptions)

• Here is the solution, but it actually is wrong/incomplete/answers a different question

By this time most new readers will have lost interest

• More stuff

The right way

• This is the standard problem (vS/Whitaker + usual (K&W) assumptions)

• Here is a simple solution.

• This is why the answer is 2/3 and not 1/2

• This is why it matters that the host knows where the car is.

So far the article is simple and interesting. Those who wish can read further

• Some sources say that the above solutions are wrong/incomplete/answer a different question

• This is why they think that the above solution actually is wrong/incomplete/answers a different question

• These are the solutions that they prefer

• More stuff.

We make no judgement at all as to which solutions are 'right' but we give both in the way that helps our readers understand the subject. Martin Hogbin (talk) 15:18, 11 April 2012 (UTC)

My problem with this is it DOES make a judgment as to which solutions are 'right', in particular is says the "simple solution" is right (why else does the article lead with this solution?). Contrast this approach with the approach I'm suggesting above (which DOES NOT make any such judgment).

• This is the standard problem (vS/Whitaker + usual (K&W) assumptions)

• Here is a related problem and its easily graspable, intuitive, simple solution

• Here is why the 1/3:2/3 answer to the simple problem must be the same as the answer to the standard problem (could include why the answer is not 1/2:1/2 as well in this section)

• Here are a variety of solutions to the standard problem (simple, conditional, game theory)

• More stuff (possibly including the controversy over the validity of the "simple solutions" to the standard problem in a much later section)

Unlike the structure you're suggesting, this structure (based on the approach used in a standard probability textbook and exactly echoed in another highly reliable source) accomplishes the same goal of helping the reader understand the subject without taking any stance whatsoever on whether any of the solutions to the standard problem are "more" correct than any others (because it presents them all equally). In the structure you're suggesting, the conditional and game theory approaches (you mean to include a game theory approach someplace, right?) are presented as if they are the ones that are controversial, where in reality the simple solutions are (somewhat) controversial and the conditional (and game theory) solutions are completely undisputed (indeed, both Selvin and Devlin follow up their own "simple solutions" with unassailably correct conditional solutions). Your oft-repeated assertion that the structure you prefer makes no judgment as to which solutions are "right" is incorrect. It definitely does. And the judgment it makes is completely inverted from what the most reliable sources we can find actually say (some of which dispute the validity of the simple solutions while none dispute the validity of the conditional solutions).

Please read WP:STRUCTURE again (Segregation of text ... may also create an apparent hierarchy of fact where details in the main passage appear "true" and "undisputed", whereas other, segregated material is deemed "controversial", and therefore more likely to be false). Also (from WP:TECHNICAL) it may be helpful to compare with a standard reference work in the particular technical field to which the subject of the article belongs - like, say, Grinstead&Snell. -- Rick Block (talk) 20:14, 11 April 2012 (UTC)

@Martin: Some people think the earth is flat, and happily no-one consider them to be reliable sources. Of course is what you call "The wrong way" the wrong way. Neither me nor Rick would support this. But what you call "The right way" isn't the right way. Here is the right way:

• This is the standard problem (vS/Whitaker + usual (K&W) assumptions)
• Here is the (correct) solution.
• This is why the answer is 2/3 and not 1/2
• This is why it matters that the host knows where the car is.

"So far the article is simple and interesting. Those who wish can read further

• Some sources present a simple solution, but this is incomplete, it is an answer to a different question
• Most maybe all presented simulations make the same mistake.
• More stuff.

We just quote reliable sources and do not make judgements of our own.

Got it? Nijdam (talk) 20:17, 11 April 2012 (UTC)

• I agree with you about the simulations. Speaking from professional experience using stochastic modeling, I believe the main reason people find them so convincing is because they give no thought to the assumptions embodied in the models, apparently believing that "garbage in, garbage out" does not apply to model design. For MHP, I have always thought it absurd that people even consider stochastic models when the modeled problem space is small enough to completely enumerate on a 3x5 card. I assume it is because enumerating it forces you to examine the assumptions, about which they feel some doubts, but faith in the almighty computer allows them to ignore the existence of assumptions embodied in the model.

  Do you have citable sources for asserting the bogosity of the models or, more charitably, for clarifying which interpretations of MHP they actually model? ~ Ningauble (talk) 00:10, 12 April 2012 (UTC)

Right, but there are of course simulations which you perform yourself (in fact you play the game), whereby one experiences the necessary assumptions. About the enumerations, I'm interested how you would enumerate the possibilities of the MHP. Nijdam (talk) 06:54, 12 April 2012 (UTC)

Regarding citable sources asserting the bogosity of the models, the (much maligned on this page and particularly detested by Martin) Morgan et al. (cited in the article) has this to say:

Solution F3: Play the game a few hundred times with the "host" using three cards: two jokers for the goats and an ace for the car. This will verify that the player who switches wins 2/3 of the time.

Several people, frustrated by contradictory arguments or failing to believe their arguments wrong, suggested schemes like F3 to settle the issue, which was proposed by vos Savant in the December article (compare, also, the classroom experiment proposed by vos Savant in the February column). It is so appealing because it models F1 [Richard's Proposition 1]. This is a correct simulation for the unconditional problem, but not for the conditional problem. The correct simulation for the conditional problem is of course to examine only those trials where door 3 is opened by the host. The modeling of conditional probabilities through repeated experimentation can be a difficult concept for the novice, for whom the careful thinking through of this situation can be of considerable benefit.

— Morgan et al.

This analysis is one of 6 of what they call the most appealing "false solutions". -- Rick Block (talk) 15:44, 12 April 2012 (UTC)

In the same journal, however, was Seymann's criticism of this paper which Ninguable has quoted above.

Later, Morgan et al, say, '...had we adopted conditions implicit in the problem, the answer is 2/3, period.' They also say that they should also have pursued the effect of the player having observed previous plays of the game. In other words the exact mathematical model of the game is a matter of choice not fact.

But none of this matters. Although I believe that the Morgan solutions refer only to an academic extension of the problem I am proposing a genuine compromise in which both solutions are fully and openly presented. Martin Hogbin (talk) 16:27, 12 April 2012 (UTC)

Structure

Rick quotes from WP:Structure, Segregation of text ... may also create an apparent hierarchy of fact where details in the main passage appear "true" and "undisputed", whereas other, segregated material is deemed "controversial", and therefore more likely to be false.

The point is that we clearly state the criticism of the simple solutions by certain sources, giving the reasons for this criticism, thus the order of the two sections is irrelevant to the argument about which one is 'right'. On the other hand it is very relevant to the probability that our readers will understand what we are talking about.

Most physics text books and all practically all mathematics books start by ignoring some of the complexities of the subject; they have to otherwise they would be impossible to understand. Books on electromagnetism often start with Coulomb's law for example, but this is in fact wrong, it only applies to the non-exstent idealised case where nothing ever moves. I could give hundreds of examples of this approach. Subject have to introduced slowly bit-by-bit or they are incomprehensible. I defy anyone to find me a maths book or encyclopedia article that does not gloss over some subtlety at the start in order to get started.

Even if I were an avid Morgan supporter I would want to do things in this way. It gives us a chance to discuss the issues raised by Morgan et al in a proper and scholarly way without making the article inaccessible to the general reader.

The only alternative that I can see is endless argument that 'Morgan are right', 'Morgan are wrong', while the article languishes in limbo.Martin Hogbin (talk) 16:51, 12 April 2012 (UTC)

You say this, but yet you're apparently opposed to the structure I'm suggesting that does the very thing you're talking about (starts with solving a simpler problem). I'm confused. -- Rick Block (talk) 19:50, 12 April 2012 (UTC)

The simpler problem is an artificial construction whose main purpose is to explain the Morgan solution. That is why it is used in G&S for example, which is aimed at people wishing to learn more seriously about probability. Many of our readers will not be interested in doing that. I have no objection to mentioning the simpler problem in the introduction to the Morgan solutions. Morgan themselves make exactly the same point, because it helps to clarify their reasoning.

We need to start with the MHP, as it is generally understood, and then explain why the answer is 2/3 and why the host's knowledge matters because those are the two points which puzzle most people. If you are in any doubt about that just look through the talk page history. We get a regular stream of newcomers claiming that the answer is 1/2 and that it cannot possible matter what the host knows. Surely that gives us a clue as to what most of our readership want from this article.

Just to clarify again, my compromise is to give the Morgan solutions a level of prominence equal to that of the simple solutions (even though I think there are actually an academic backwater) but not to give them at the start of the article. Martin Hogbin (talk) 22:52, 12 April 2012 (UTC)

You keep saying you're compromising when what you're actually doing is simply insisting that you get your way. The point of my suggestion is NOT to explain the "Morgan solution", but to explicitly put the reader in a frame of mind where your beloved "simple solution" actually becomes intuitive to most people. The regular stream of newcomers claiming the answer is 1/2 has not stopped, even though the structure of the article has been what you are arguing for, for approximately a year now. Surely that gives us a clue that this structure (the one you are insisting on) does not provide the benefit you claim you want (making the answer easily understood). My suggestion is a NEW idea - fundamentally unrelated to whatever disagreements we might have had in the past about the validity of this or that solution. Frankly, I'm past that. It would appear to me that you're not. -- Rick Block (talk) 00:14, 13 April 2012 (UTC)

The current structure is not what I am suggesting it is now a complete mess. Martin Hogbin (talk) 08:37, 13 April 2012 (UTC)

I have a challenge for you. Pose the Monty Hall problem (using whatever version you'd like) to anyone you'd like who's over the age of 12. Before they answer, ask a few more questions (in this order). 1) What is the probability the car is behind door 1 BEFORE the host opens a door? 2) So if you decide you're going to stick with your original choice no matter what the host does, what is the probability you'll win the car? 3) What is the probability there's a goat behind door 1 BEFORE the host opens a door? 4) So if you decide you're going to switch to whichever door the host opens (i.e. no matter what the host does), what is the probability you'll win the car? My hunch is nearly everyone gets all of these right. Establishing the right context is the key. -- Rick Block (talk) 00:26, 13 April 2012 (UTC)

Lest you think I'm kidding, tonight I actually did what I'm suggesting you do. 3 out of 3 (fairly random adults - and I understand this is not a statistically significant sample size) got the first 3 questions correct. They had more trouble with the 4th, but with (minimal) more explanation were able to grasp that the answer to the 4th (however unintuitive it may be) must be the same as the answer to the 3rd (the probabilistic complement of the answer to the 2nd). Just FYI. -- Rick Block (talk) 06:42, 13 April 2012 (UTC)

We all know that it is the fact that the host opens a door to reveal a goat that makes the problem unintuitive. We all know that the player makes their choice after they see a door opened. The disputed point is whether it matters which door the host opens. Your challenge is an irrelevance as is your own OR on the subject. Martin Hogbin (talk) 08:37, 13 April 2012 (UTC)

Like you, I really would appreciate some input from others, although you are welcome to comment too. Does anyone else see the point of starting simple? Martin Hogbin (talk) 08:37, 13 April 2012 (UTC)

Section 2.6 ("Alternative derivations")

The last sentence, "The conditional probability is ... 2/3," is ambiguous (which conditional probability?). I originally interpreted it as the conditional probability of the host opening Door 3 (the most recent one mentioned), and confused myself because that probability is obviously 1/2.

Also, the section appears to be copied from Gill's paper with a few modifications. The modifications, incidentally, make the section much less clear; the corresponding section in Gill's paper is unambiguous, saying that "the conditional probability of winning by switching ... equals the unconditional probability of winning by switching, 2/3." I'm not sure what to do here: I don't know if I should just fix the problem, or remove the section entirely for being a copyright violation. Eyu100^{(t|fr|Version 1.0 Editorial Team)} 01:05, 13 April 2012 (UTC)