User talk:Gerhardvalentin/MHP
"What is the conditional Pws" vs. "Should the contestant stick or switch?"
[edit]To whom the question is addressed to?
Evidently to the person who should answer that question, let's call it the ADDRESSEE (A).
A: A is free then to answer the question based on the perspective/knowledge of the contestant C, or to answer the question based on his/her/some own/other/assumed perspective O.
C: If A chooses to use perspective of C as the basis of his/her answer, he/she should consider what C could eventually know resp. cannot know.
O: If he/she chooses the perspective of O he/she obviously is free to "define" his/her own state of knowledge in a reasonable way freely, i.e. to take/define a variety of own assumptions/axioms. Just necessary to be clear and consistent, not leaving doubts / open questions.
Generally accepted: O.
Necessary in any case to fix the:
Underlying conditions
[edit]Generally accepted:
- When making her first selection, the contestant has no knowledge about the actual distribution of the three objects.
- The host shows a goat and offers to switch in any case, e.g. not only if the contestant has selected the car.
- If the host has two goats to choose from, he will not give away any further information about the actual location of the car, thus not opening one special door above average nor whenever he can.
Marilyn vos Savant clearly expected her readers to assume that the game was played in this manner.
The insertion and the use of absurd "additionally given information" and far-fetched other insertions and assumptions, like some "Host's bias etc., that some mathematicians and probability theorists like to over-analyze and to read into the famous question has to be rejected for this reason, never to be imputed in this world famous paradox. Such complications serve for purpose of exercises for students of probability theory only, but cannot affect the famous paradox in any way.
Other, less reasonable conditions / formulations based on Whitakers 1990-question
[edit]Should she stick or switch
[edit]- What, if the Organizer / the show-team "always" are hiding the car behind door 1, and the contestant knowing about that bias?
- What, if everyone in the audience of course knows about the car always being behind door 1, but just the contestant has no knowledge at all about the actual distribution of the three objects?
- What, if the Host always choses a special door, say No. 3 to open, if ever possible, and 95 % of the audience knows about? And what if just the contestant doesn't?
- What, if the Host always choses a special door, say No. 2 to open, if ever possible, and no one knows about?
- And, such bias being given, what if everybody had known that already long before, including the contestant?
- The "lazy Host", the "drunken host"
- The "ignorant Host" etc. etc. etc.
- ad 1+2): If the car always is located behind the same door (regardless of the contestant's knowledge), and A (the addressee of the question "stick or switch") definitely knows about or just (O) is "considering such possibility", A has to answer this question based on his/her knowledge/assumption, giving clearly account on such knowledge/assumption.
- 3-5): Whitaker's 1990-question does not imply nor provide for any such guaranteed "host's telltale bias", always giving away closer information on the actual location of the car. Nevertheless A is free to – and obviously A even should consider the possibility of such a potential Host's bias. He/she never can take such an ultimately insincere bias to be granted/guaranteed, but – in his/her answer – should address explicitly consideration on similar less important side scenes. Showing that the most extreme "Host's telltale bias" will lead to the very least Pws of 1/2 (ie 2/3 minus 1/6) twice often than to the maximum PWS of 1 (ie 2/3 plus 2/6), on average inevitably always commuting around 2/3.
- 6-7 see 3-5): Even such absurd ideas and less important side scenes may briefly be discussed at the end of the article.
Such and similar unreasonable and ultimately insincere bias-suppositions and formulations may be rather interesting, entertaining and amusing for teachers and students of probability theory, but they never do address Whitackers question, and they never are addressing or elucidating the world famous 50:50 paradoxon of the MHP.
It is fully recognized that – before opening of a door – just to know the host "will very soon be opening" one of his two doors, is no new condition. Coherently: In Whitacker's question, no one knows about any "host's bias". And therefore it is irrelevant which one of his two doors he will be going to open. Although conditionalists – already knowing about some host's bias – they anyway in this stage could present their findings: "In case he will open his favored/unfavored door 2: then so-and-so". But: "if he will open door 3: then so-and-so, because if he opens his unfavored and avoided door, then his favored door is likely to hide the car". They could do that, not even needing abstract probability theory. But they don't. They say: Before he has opened "his" door, without knowing the host's "bias", we cannot say anything. But after the host has opened "his" door, still being in the dark and nothing knowing about any host's "bias", they suddenly are affirming to have found a "new condition/important new information", and by applying "indispensable conditional probability theory" then, all they can show you indeed is "which door" the host has opened, nothing more. But you already know that without conditional probability theory.
Solely having shown one goat, by opening of one of his two doors by the host never can count as a "new condition" for the MHP. A "new condition" can only arise under the supposition that, for whose perspective ever, it will be "casting some insincere additional light" on the actual location of the car, by some existing and known bias of the host e.g., otherwise not. Securing for each and every game that “we’ve learned absolutely nothing at all to allow us to revise the odds on the door initially chosen by the guest (Falk)”. – But, already knowing about or expressly assuming such a bias, you never need conditional probability theory to get alledged "better advice" for the decision to switch. Gerhardvalentin (talk) 14:49, 23 August 2010 (UTC)
Copied from Citizendium MHP:
[edit]For some readers, numbers speak louder than words. The following table should be self-explanatory.
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We observe that the player who switches wins the car 2/3 of the time. We also see that Door 3 is opened by the host 1/2 = 1/6+1/3 of the time (row 2 plus row 3), as must also be the case by the symmetry of the problem w.r.t. the door numbers - either Door 2 or Door 3 must be opened and the chance of each must be the same, by symmetry.
Winning by switching in combination with Door 3 being opened occurs 1/3 of the time (row 3). The conditional probability of winning by switching, given Door 3 is opened, is therefore (1/3)/(1/2)=2/3. Since this is the same as the overall chance 2/3 of winning by switching, we see that knowing the identity of the opened door doesn't change the chance of winning by switching. Not only does the switcher win 2/3 of the time, he also wins 2/3 of the time that Door 3 is opened by the host, and 2/3 of the time that Door 2 is opened by the host.
To say the same thing in other words, the combined chance of winning by switching and Door 3 (rather than Door 2) being opened, 1/3, equals the product of the separate chances of "the car being behind the other door", 2/3, and "host opens Door 3", 1/2. Whether or not the car is behind the other door to the door opened by the host is statistically independent of whether the host opens Door 2 or Door 3.
This last fact could have predicted in advance, by the symmetry of the problem. The contestant might as well ignore the door numbers: they don't change his chances of winning by staying or by switching.
- This was copied from Citizendium. Gerhardvalentin (talk) 20:21, 7 February 2011 (UTC)
Conditioning on door numbers
[edit]Conditioning on door numbers is a useful and an applied method in teaching and training conditional probability maths. See adequate text books to the purpose (available en masse). But there is no need to condition on door numbers in deciding whether to stay or to switch in the situation of that one-time show, painted by MvS, that never ever happened in reality, never in exactly that specified appearance. MvS unfolds a clean paradox. MvS never told that the host commented in this one-time show "I did not like to open that special door that I have opened, I opened it only because my second still closed door hides the prize". MvS would have been misinterpreted if s.o. tells that she had said so. She didn't. You don't know the host's background, the guest sees him for the first time.
It is an imaginary show, with an imaginary host. No one has ever seen him in reality. You have no reason to "assume" that he might be giving away the prize just "for free" by remarkably facilitating your chance to win by any special behaviour, in this imaginary one-time show, by NOT observing secrecy regarding which door hides the prize. Why should he? You cannot assign anything to his opening of a door. Even if you have been told that it "could have been door 3", e.g. So you cannot infer from it any difference to his opening of "door 2", e.g. No difference at all. But you are free to "assume" that he could have told that his second door hides the price. But this will be without any retroaction to reality.
Yes, some left the one-time show without saying so and said that his choice of door (if he should dispose of two goats in 1/3) could be giving away some hint on his very special bias to eventually prefer one special door. This is mathematically fully correct. Forgetting however that this could be the case only if he indeed HAS such bias and that you definitely KNOW about that bias (Ruma Falk). Once more: that he has some bias and that you know about this bias, its extent and its direction. Yes, this is the necessary principle that all those textbooks on conditional probability theory are based on. Such knowledge about such bias could indicate and could be giving away some additional information just for free, regarding the actual location of the car behind one of the two still closed doors. This is a profitable goal only in training conditional probability theory. But it is fully obvious for any observer that only God may be able to interpret the essential being of a totally unknown host and his possible bias, and if so its extent and its direction, but absolutely no-one else (Tom Hulse: It's relvant to God if he was wanting to compute probabilities with all secret thoughts included in the formula). So it is absolutely impossible for an unknown host to give away such hint in a ONE-TIME show. Absolutely impossible. His absolute *inability to blab* has correctly been translated as follows: "The host is observing *secrecy* regarding the car-hiding door" (N.Henze), or "The host *chooses at random* which door to open if both hide goats" (K&W). Expressing that it is absolutely useless to condition on which of his doors the host just has opened resp. he left closed.
So anyone knows that, in that ONE-TIME show with an unknown host as per MvS, conditioning on door numbers is futile and unnecessary. And everyone knows for sure is that he never will be able to signalize in honesty that staying ever could be the better choice for you to win the price in this special game, because "always switching" must give you the car with probability 2/3, and "always staying" can give the car with probability 1/3 ( ! ) only, so any "staying" will damage your chance to win, therefore staying a priori is forever excluded. Just from the outset, forever excluded. You clearly see that textbooks rightly do condition on door numbers, but in striving for quite another goal.
For finding the correct answer in the paradoxical situation the contestant is in, conditioning on door numbers is "needed like a hole in the head." Gerhardvalentin (talk) 23:34, 8 October 2012 (UTC)
- (transferred my comments from Martin's talk page Gerhardvalentin (talk) 11:25, 9 October 2012 (UTC)
Richard said on his [talk page] on 10 October 2012:
- Good, that's clear. For a given specification of the quizmaster's strategy there is a best strategy of the player. But we don't know it, are not told about it. Even if we observed the game many times in the past, how do we know that the host will behave the same way, tonight? All we know are the rules: the host will certainly show us a goat behind another door. "Randomize and switch" is the unique minimax strategy: gives us the best chance of winning, whatever the host does. Assuming the host doesn't want us to win, his minimax strategy is: hide the car completely at random and open a different goat door completely at random. Symmetry is at work here, too. The problem is symmetric in the door numbers. A minimax solution exists, by von Neumann's theorem. By symmetry, there exists a symmetric minimax solution. All this known since Nalebuff popularized MHP in the decision theory literature soon after Selvin did in statistics, long before Vos Savant made it famous in popular literature.
And I say:
- "Assuming asymmetry" in the host's choice (if he got two goats in 1/3) is a weird assumption and could be compared to his just opening of all three doors at once, before offering to switch to one of his two doors, like exceptionally opening his strictly avoided door. Asymmetry runs counter to the host's interest. His only lifesaver is absolute symmetry. Besides: in that "one time" show the question is about, the host is totally "incapable to show any bias". Henze calls this very fact "secrecy".
So everything has to base on symmetry, "door number opened" is completely irrelevant, therefore. Once and for all completely irrelevant to give the correct answer. Disused. The article especially also should cite modern sources, and the article should clearly show and say that door numbers / door location once and forever is irrelevant. To condition on door numbers, hoping to get a "more precise value" than 2/3 is viewless and never needed to decode the paradox nor to give the only correct answer.
Conditioning on door numbers saying it gives a "better value" is a viewless very narrow mathematical field of view, irrelevant for giving the correct answer. Conditioning on door numbers is rewarding solely in training conditional probability theory. See all those textbooks. We should cite modern sources, and the article should say so. Gerhardvalentin (talk) 19:57, 10 October 2012 (UTC)