Revision as of 09:58, 1 January 2011

Template:Editnotices/Group/Wikipedia talk:Requests for mediation

Front matter

Archives

• Archive 1
• Archive 2
• Archive 3 - Opening statements; discussion of issues
• Archive 4 - List of issues; Whitaker, Carlton
• Archive 5 - Questions posed by Sunray; Work on conditional solution section
• Archive 6

Subpages

• Conditional probability solution

Groundrules

In most of the successful mediations I've been involved with, there is a transition from dispute to collaboration. The best mediations involve learning. One thing we can all learn more about is how to navigate through the maze of WP policies. It is usually beneficial to agree on some "first principles" that will promote a shift to collaborative editing. In my opinion it will improve our chances of a successful outcome if all participants sign their agreement to the following principles:

• Focus on content rather than the contributor. Note: This is to be interpreted literally, as worded.
• Be guided by WP content policies, particularly WP:V and WP:NPOV
• Commit to being as economical as possible in posts to this discussion page.
• Work towards consensus in editorial decisions.

Easy right? If you have any questions or comments, by all means start a new section below the signatures. Please signify your agreement below:

General discussions

Misplaced Pages talk:Requests for mediation/Monty Hall problem/General discussion

Proposed compromises

Misplaced Pages talk:Requests for mediation/Monty Hall problem/Compromises

Merry Christmas

I'll be away for some days, only to return just before Christmas. I take this opportunity to wish you all, AGK, Gerhardvalentin, Glkanter, glopk, Kmhkmh, Martin Hogbin, Richard Gill, Rick Block, Sunray, a Merry Christmas and the best wishes for the coming year. Nijdam (talk) 22:43, 15 December 2010 (UTC)

I would also like to wish everyone a happy Christmas and best wishes for the coming year. Here is my Christmas present, it's for Nijdam and Rick. Rick said that he thought that the reason you must compute a conditional probability is because of the moment when the decision has to be made to change. Nijdam also has this point of view: the conditional viewpoint corresponds to the player who makes his decision after he has seen Door 3 opened. However, probability theory is not normative, it does not prescribe what you *must* do. However, there are some very good reasons why it is *rational* to compute the conditional probability "at that moment of time". Here they are: I will give two reasons, one for each of the main streams of thought:

Subjectivist: for the subjectivist, information comes in and beliefs about the state of the world are continually updated. If we have modelled the MHP as a four stage process going on in time (car hidden, door chosen, door opened, reconsideration of choice) and if we are a rational subjectivist then our beliefs are updated in each phase: car hidden -> 1/3,1/3,1/3 -> door 1 chosen -> 1/3,1/3,1/3 -> door 3 opened -> 2/3,1/3,0. Subjectivist probability theory is a theory of how a "rational person" *must* adjust his degrees of belief about unknowns as new information is received. Our rational degree of belief at any stage is *the* conditional probability distrbution given *all* that we know at each stage.

Objectivist: for the objectivist, there are no normative rules coming from probability theory. But in a decision problem like MHP it is agreed that the problem is solved by finding the *optimal* strategy, and by *proving* that it is optimal. The overall probability of success is the target quantity which we want to optimize. The simple arguments of the simple solutions tell you that always switching has overall (unconditional) success probability 2/3. We have not finished "solving" MHP if we haven't mathematically proven that 2/3 can't be beaten. The only way it could be beaten, would be if there was some situation recognisable to the player (door chosen, door opened) in which the conditional probability that switching gave the car was strictly less than 1/2. This follows easily by consideration of the law of total probability: for any strategy whatsoever, the overall success chance equals the sum over the six possible situations of the probability of each situation multiplied by the conditional probability of success with that strategy, given you are in that situation.

I would suggest to the conditionalists that they search for reliable sources which give these arguments, in order to make their case stronger. The reliable sources from probability theory and statistics which write on MHP don't give these reasons explicitly, because their intended audience already knows them, often instinctively rather than explicitly; and also because sources usually try to express themselves "probability interpretation free", in order not to get into heated controversy. Richard Gill (talk) 04:36, 18 December 2010 (UTC)

Thanks Richard, for your good wishes and Xmas present. I especially invite Martin, Gerhard and G. to study it. I also want to emphasize that the solutions presented in the sources, and referred by us (me cs.) as "simple solutions", are NOT the solution using the symmetry to calculate the conditional probability. See for instance what Devlin writes as solution, or the solution known as the combined door solution. These solutions are mathematically wrong, and hence have to be criticized. Nijdam (talk) 16:14, 30 December 2010 (UTC)

I hope you had a good Xmas Nijdam. All solutions, including the simple solutions and the conditional solutions in the article, can be subject to criticism. The point is, when and how do we criticise them? My suggestion is the we present the solutions (in particular the simple solution) first, complete with explanation, and later on we criticise them in a sober and scholarly manner. The reason for this is simple. Most people who come to the article will find the hardest thing to understand and believe is that the answer is 2/3 and not 1/2, regardless of whether the problem is conditional or not. First we should explain why this is so.

I have used the example of good text books (and encyclopedia articles) before to explain this approach. It is how almost all such sources do things. Simple, maybe glossing over some details, first, then a more in-depth discussion. You have never explained what you have against this approach, Nijdam. Martin Hogbin (talk) 17:20, 30 December 2010 (UTC)

I also like a "simple explanation", as long as it mentions something about the probability before and after the opening of a door by the host. I.e. it could mention that the initial probability for door No. 1 to hide the car is not influenced and hence the probability after door No. 3 is opened also is 1/3. That would be correct reasoning, other than the simple solution does. Nijdam (talk) 17:42, 30 December 2010 (UTC)

Why not? So long as we can do this in a way that does not draw unnecessary attention to it in a way that might put off non-expert readers. Why do you not propose something that could be added to the simple solution. Not to make or prove a point but just to make it more correct. Martin Hogbin (talk) 20:13, 30 December 2010 (UTC)

Marilyn vos Savant's original simple solution was, as she explicitly confirmed later, from the outset based on a host who, in opening of a door, would certainly not give away any closer hint on the actual location of the car. So, by the host's opening of a door, we've learned absolutely nothing to let us revise the odds on the door originally selected by the guest (1/3). In this version they remain unchanged. As much to the odds on the car originally selected by the guest, before and after the opening of a door by the host. Independent of the door No. Gerhardvalentin (talk) 23:08, 30 December 2010 (UTC)

There are several sources which make this same point. Martin Hogbin (talk) 23:20, 30 December 2010 (UTC)

Well, how can opening of a door with a goat, not tell anything about the location of the car? I hope I do not have to explain this further.Nijdam (talk) 08:52, 31 December 2010 (UTC)

Nijdam, you are just being awkward now. It is quite obvious that showing a goat behind door 3 tells us the car is not behind door 3. The point, that you make yourself above, is that (in the symmetrical case) this does not alter the probability that the car is behind door 1. How would you wish to make his point (as you propose above) in the article? Martin Hogbin (talk) 10:15, 31 December 2010 (UTC)

Who is awkward here? Yes, the car is not behind door 3, yet the probability the car is there is, due to the random placement, 1/3!! Got it?Nijdam (talk) 11:00, 31 December 2010 (UTC)

You continue to quibble about a simple mistake but make no attempt to engage in cooperative editing. You yourself say that, in the symmetrical case, 'initial probability for door No. 1 to hide the car is not influenced' yet you will not discuss how we might add something to that effect to the article in a way that might be acceptable to everyone. Martin Hogbin (talk) 11:06, 31 December 2010 (UTC)

Sorry Martin, I'm definitely willing to formulate a correct "simple explanation". But all in its proper time. For the moment I do not understand what you refer to as "simple mistake". My efforts here still are mainly to make you, and I hope Gerard too, understand why the simple solution is wrong, because of the lack of the right arguments.Nijdam (talk) 00:59, 1 January 2011 (UTC)

The simple solution gives a clear argument for the correct answer to the MHP-question "is it to your advantage to switch?", i.e. for the decision asked for: "YES or NO". –  The answer clearly has to be YES, and the decision is SWITCH, for staying never can be better. –  Any correct conditional approach gives a "correct conditional result", but no correct conditional result has ever been able to proof the contrary, to proof that staying could ever be a better decision, in any specific game-show, if there should have been more than one.

So it is more important to criticize the illusion that, under the reasonable assumptions, a "conditional probability" is needed as a "better basis" for the decision asked for. And that a conditional approach really is indispensably necessary as a "better" basis for that decision. Such illusions are missing the point and therefore fail to heed. And so I somehow agree to Martin and to W.Nijdam. Regards, Gerhardvalentin (talk) 00:23, 31 December 2010 (UTC)

I don't understand what you mean to say here. The simple solution says "switch", but based on the wrong argument. And it does not say: "for staying never can be better", whatever that should mean. And then, why should a conditional approach prove that staying could ever be a better decision? Where is your logic? Nijdam (talk) 09:05, 31 December 2010 (UTC)

W. Nijdam: The argument of the simple solution never is that it is guaranteed to win by switching. Never guaranteed.

>>No one said so, so why do you mention this?

The simple solution just says: Pws is (only) 2/3, on the long run.

>>No such thing as "on the long run". You confuse probability with its interpretation.

That's enough.

>>Enough for what? Certainly not enough to base your decision on.

Therefore it cannot ever be of any advantage to stay.

>>In the specific situation you do not yet know this.

Based on this argument you never should stay, but always should switch.

>>The word never is not appropriate here. And the argument fails.

And this argument is confirmed to be correct by any correct conditional approach.

>>No, the conditional approach gives the correct argumentation, the simple explanation does not.Nijdam (talk) 15:19, 31 December 2010 (UTC)

Even any tattletale host is out of position to make staying more advantageous, even if you should (but you cannot!) "know" the exact conditional probability in any single act. Even any tattletale host cannot avoid that Pws will be 2/3, on the long run. Never. That proves the argument of the simple solution to be correct, and no conditional approach can ever contradict this implication of the simple solution. You always should switch, and staying never can be "of any advantage". Never. Absolutely never.  Gerhardvalentin (talk) 11:13, 31 December 2010 (UTC)

Gerhard, I have to agree with Wietze here. Indeed, the simple solution tells us that "always switching" has an overall success chance of 2/3. inspection of the conditional probabilities shows us that this couldn't be improved by sometimes switching, sometimes not, depending on the configuration numbervof door chose, number of door closed. The simple solution only compares "always switching" with "always staying". But there are other possible strategies. BTW, by "simple sokution" I mean any correct argument that always switching has success probability 2/3. I think that by "simple solution", Wietze usually refers to some incorrect argument. Fortunately he does now agree that there exist mathematically correct proofs that "always switching" has overall success probability 2/3. Richard Gill (talk) 16:29, 31 December 2010 (UTC)

I have to correct Richard here: I never was of a different opinion. There are several proofs concerning the overall probability, being 1/3 under the right assumptions. Like Morgan, I consider this a correct solution to a slightly different problem, but not appropriate for the usual MHP. Nijdam (talk) 00:41, 1 January 2011 (UTC)

A Different Tack - Is The 50/50 Host A Premise Of The MHP?

The reason I ask, is because I feel there is undue weight given to the issue of the host *not* picking uniformly between 2 goats.

I would consider the article a lot closer to NPOV if all the references and discussion of host behaviors that are *not* 50/50 were removed from the Conditional solution section.

My reasons for suggesting this are:

1. It is not a significant POV of the reliable sources that the MHP does not include the 50/50 premise
2. Without the 50/50 premise, the answer can not be declared as 2/3 & 1/3, according to the reliable sources
           But it will be 2/3 & 1/3 forever, on the long run, even without the 50/50 premise, even with any most extreme bias!
           It just can matter in "one isolated single special game"! – But never on the long run. Never.   Gerhardvalentin (talk) 11:38, 31 December 2010 (UTC)

The switcher gets the car with unconditional probability 2/3 if and only if the stayer gets the car with unconditional probability 1/3. Host bias is totally irrelevant to this, the simple solution, at least, to simple arguments for the simple solution. However, if you want to have conditional probability 2/3 given specific choice of door and door opened, then you have to assume no host bias, or, if you are a subjectivist, neutrality with respect to host bias. But anyway, as long as initially all doors are equally likely to hide the car, all conditional probabilities are at least 1/2 and they average out at the unconditional 2/3 (law of total probability). So host bias is pretty irrelevant to both solutions.I thought Glkanter had no use for conditional probability so I don't see why he has any interest in host bias. Gerhard: if you want to think about long runs, please distinguish the overall long run, from each of the six subsets defined by initial door choice and door opened. Unconditional and conditional is about overall versus one of the six special cases. Richard Gill (talk) 09:58, 1 January 2011 (UTC)

1. The Great Paradox is 'why is it 2/3 & 1/3' rather than 50/50?'
2. This is the 'Solutions section' to the MHP, not the starting point of a text book on conditional probability
3. These paragraphs (2 & 4 in the section) are used to support the opening paragraph of the section, which is the POV that the simple solutions are flawed, they do not support how the conditional solution returns the 2/3 & 1/3 result.
4. The Misplaced Pages MHP article makes it very clear that most interpretations of the problem include (assume?) the 50/50 host.
5. There are no sources who specifically state any other behaviour for the host is part of the MHP.
6. The ambiguity of vos Savant's problem statement, and subsequent differences in the reliable sources in this regard is addressed clearly earlier in the Misplaced Pages article.
7. In addition to the mention prior to the Solution sections, the host behaviour is discussed at length in numerous other sections of the article.
8. Both images that support the conditional decision tree solution in the section *do* use the 50/50 premise, just like the simple solution requires it, and just as the conditional Bayes Theorem solution solution in the article requires it.
9. Problem statements using a host bias other than 50/50 are considered 'variants' (different problems) of the MHP by the reliable sources and in the Misplaced Pages article, not the MHP itself.
10. The MHP is to be solved from the game show contestant's (Selvin & Whitaker/vos Savant) State of Knowledge. No sources ever explain how the contestant becomes aware of any such host bias.

Glkanter (talk) 08:56, 31 December 2010 (UTC)