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Thread: Mythbusters- Monty Hall 3 Door Pick

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  2. #17
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    Quote Originally Posted by Canceler View Post
    I liked what Dodo had to say about information, except if this were to happen...
    ... I'd change my guess to the door that's open that I can see has the prize behind it.

    SoFlaLaw-
    It might help to think about sagefr0g's 1000-door version of the situation. You pick one door. Of the remaining 999 doors, I show you the prize is not behind 998 of them. According to you, your chances are now 50/50 whether you change doors or not. You have to ask yourself, does this really make sense?
    Absolutely, when there were 1000 doors, my odds were 1:999. When 998 doors are removed, there are only 2 doors remaining, my selection and the other. At that point in time, it is as if the other 998 revealed doors never existed as a legitimate selection opportunity. So, at that point in time, when I am afforded the opportunity to make a switch decision, there are but the 2 doors. My odds do not remain constant at 1:999, but rather, become 1:1.

    Reverting back to the 3 door option, the game show host (assuming his/her knowledge is relevant) exposes Door 2 or Door 3 that does not have the prize. Were he/she to have knowledge as to which door the prize is behind, he/she might know it's behind Door 1 (the door I selected), and can't show it to me as that part of the show would be over too quickly. Alternatively, if the prize is not behind my door, he/she could first offer me the opportunity to switch to either of Door 2 or Door 3, and then expose my Door 1 (Interesting twist, would he/she then offer me chance to switch again?).

    Any way you slice it, once 1 door is exposed without the prize, the odds for my success improve from 1:2 to 1:1 (2 doors, 1 wins, other gets Daisy the cow or goat). I understand the passionate defense of Bayes Theorem, but I believe it flawed, as it does not take into account the removal and irrelevancy of the other options from the "new" selection decision (switch or stand 1:1).

    PS - Just as easily (let's keep with 3 doors for example), couldn't the 2:1 perceived advantage supported by Bayes Theorem work in reverse, that is, couldn't the 2 advantage opportunities could include my door and the exposed door, and the disadvantageous position of 1 opportunity be the door that I could switch to? So, in one way it would be 2:1 in my favor, and the other way, it would be 2:1 against me. Combining those two possible results yields 3:3 (advantageous results versus to disadvantageous results), which reduces to 1:1 (50%).
    Last edited by SoFlaLaw; January 5th, 2015 at 03:02 PM.

  3. #18
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    Quote Originally Posted by Katz View Post
    exactly Katz!!!, imho
    the problem isn't really a math problem so much as a mystery of the universe problem, really. a practical math problem of application perhaps but not really a math problem. so that’s probably why some for real mathematicians got the problem wrong. word problems are difficult, lol. it’s really a science problem, one that needs the scientific method applied in order to know how to apply the math. observe and gather facts, hypothesize, experiment to see if the hypothesize can be disproved, ect., ect. the scientific method, the process we’ve used to make all kinds of incredible discoveries and things happen, from landing on the moon, landing a car and driving it on mars to keeping cancer patients alive and healing broken malfunctioning hearts. but anyway, fundamentally i think the problem boils down to stupidity or ignorance and observation and intelligence or information. that’s really what the problem is all about. it’s an advantage play problem.
    after i posted my reply to SoFlaLaw, i thought (while driving to a casino) man what a bull crap reply i gave him, just regurgitated pretty much what Canceler and Aslan said is all, not that they aren’t correct. i mean heck, i can’t really tell a guy like SoFlaLaw (that knows Bayes Theorem) anything about math, what the heck do i know, nothing really just some arithmetic , geometry, algebra, calculus and differential equation class’s is all I know. i never even had any formal education regarding probability stuff excepting maybe Schrodinger’s equation we used in advanced physical chemistry for mapping out the ‘location’ of a hydrogen atoms electron, sorta thing, that class should have had a prerequisite of a probability course, which i never had . but anyway i made a vain attempt to assert the importance of the first decision in the problem where we choose our first door, but didn't really back up the significance and validity of what that meant. just regurgitated the fact that the last remaining door we didn't choose and wasn't opened 'deserved' the dumping of the now 'extra doorless' 1/3 chance onto it, so that it was now 2/3 a chance that the prize was behind it. but i'm thinking while driving down the road, well SoFlaLaw is gonna be thinking, oh really?, how so? why couldn't that 'extra doorless' 1/3 chance just be dumped onto our original door, so that it now has the 2/3 a chance that the prize was behind it? what makes this last door so special? what's wrong with the original door we chose?
    Dodo is right there is an element of collusion going on in this problem, due to the inside knowledge that monty has. thing is i don't believe the collusion factor or inside knowledge factor is absolutely needed for the problem to exist in a still fundamentally valid way. you just need for the prize to not end up being revealed before the game is over, is all, then the problem can be played out with or without collusion.
    i believe there is a far more fundamental problem hiding in this problem. it was kind of alluded to in zg's post, where he wrote about "He would know that most of the time people's first pick would be wrong because they made their pick from a larger set." well the point zg made about the wrongness of picking from a larger set is correct mathematical thinking but it is not the core of the fundamental issue either, imho, what is core here is that He would know most of the time people's first pick would be wrong for any problem of this fundamental nature. point being, just to initially participate in this problem (unless it's a free roll or you know about the second (offer) part coming) is gambling and is a really dumb thing to do. it’s still really gambling even if it is a free roll, after all monty virtually forces one to make that initial decision. an initial decision that for all intents and purposes is just a dumb gamble. i can remember how confused i used to be when i'd see people gambling in casino's or on tv or a movie. it was like, i see these mostly well dressed intelligent looking fairly wealthy people putting their money down on these various games and apparently making decisions about the game, one way or another. i just couldn't understand what on earth it was these people were thinking about, what was going on in their heads as they made these bets and game moves. it wasn't until i learned about advantage play that i understood that these folks had absolutely nothing of value or intelligence what so ever going on in their heads, they were in fact just mindlessly gambling, lol. point being, gambling is an indicator that an unwise decision had likely been made, not a wise one.
    let's change up this problem a bit. let's say monty is the devil and the goats behind the doors are his evil angels of death, the prize behind the one door is a normal life (with oh ok a million dollars as icing on the cake). so now this problem becomes what? a problem of life or DEATH! so what would any sensible person do if they even saw monty coming their way?, run like hell, get outa dodge. knowing that to even play this game essentially means sure death.
    so hopefully it's now apparent why that 'extra 1/3 chance' isn't added to the first door we chose. hey, if you made that decision to play this game (life or DEATH) you’d all of a sudden realize you just did something very, very, very dumb. that door was a just plain stupid gamble (which exists having the wrongness of picking from a larger set) a gamble that should have never been made in the first place and in no way deserves an 'extra 1/3 chance' added to it. but this last door for which we should attribute this 'extra' 1/3 chance to making it worth 2/3 chance is worthy of that addition because it can be hypothesized that to load the extra chance onto that door is at least not putting it onto an option that was stupid to make in the first place. we put that extra chance onto a choice that has not been sullied by stupidity but which has been adorned with intelligence, information and observation. for the life or DEATH version of this game we choose the door which gives us a better chance to live after having made a stupid choice that lead to the disastrous ignorant high chance of death. Thus we use the scientific method, not just mindless gambling, hence we assign that 1/3 chance detached from the goat bearing door to the unsullied door remaining. we are not throwing good value after bad value when we hold back the ‘extra’ 1/3 chance value from our original door.
    the finally really most interesting thing about this problem is imho how value of expectation is variable dependent upon the nature of how the constructs of the problem are observed. for example, a person playing the game from beginning to end should conclude the unperturbed door has a 2/3 chance of holding the prize while a late comer to the game would conclude the two remaining doors hold a fifty fifty chance of holding the prize. this in a way reminiscent of how nature appears dependent upon observation in cases such as the quantum split slit dual nature particle/wave experiment and Shrodinger’s alive/dead cat thang.
    best regards,
    mr fr0g MMOA honorary predator
    STRENGTH - HONOR - HEART
    that's my take on it your mileage may vary.
    for senior citizen fuzzy count click link:
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  4. #19
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    sagefr0g,

    Thanks for the response. I guess that I will just agree to disagree. Your life or death example applies the 1/3 probability to the remaining non-selected door based upon a "values" (bad decision - do not reward the ploppy for playing the game in the first place with Mephisto, but reward the guy/gal for using their brain in trying to save their life now that it is at stake by applying the 1/3 probability to the non-selected door) judgment.

    To me, either way, the 1/3 probability from the exposed door can be applied to the initially selected door or the non-remaining door, thus I would assign 1/6 to each door (splitting that 1/3 probability), resulting in a 1:1 or 50% versus 50% probability. I do not care whether the initial decision was based on dumb decisionmaking or greed, or if the later decision was based upon a sincere effort to save a human soul (and win $1million to boot, or the devil's golden fiddle - ask Charlie Daniels Band if you do not understand this reference). Emotions and values judgments are irrelevant to my attempts to analyze this problem.

    Thanks for trying to shed light on my inability to grasp how this works according to most APs' explanations I have seen online since starting to CC.

    I appreciate the civil tone that all have thus far displayed regarding my inquiries.

    HNY

  5. #20

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    Quote Originally Posted by SoFlaLaw View Post
    To me, either way, the 1/3 probability from the exposed door can be applied to the initially selected door or the non-remaining door...
    Not exactly. It's dynamic in that we SWITCH in order to "change" the odds.

    A = 1/3
    B = 1/3
    -------
    C = 1/3

    I choose C
    A+B = 2/3
    Eliminate A...
    ... and B = 2/3 (so SWITCH)

    From Wikipedia: http://en.wikipedia.org/wiki/Monty_Hall_problem

    Many readers of vos Savant's column refused to believe switching is beneficial despite her explanation. After the problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine, most of them claiming vos Savant was wrong (Tierney 1991). Even when given explanations, simulations, and formal mathematical proofs, many people still do not accept that switching is the best strategy (vos Savant 1991a). Paul Erdős, one of the most prolific mathematicians in history, remained unconvinced until he was shown a computer simulation confirming the predicted result (Vazsonyi 1999).

    The problem is a paradox of the veridical type, because the correct result (you should switch doors) is so counterintuitive it can seem absurd, but is nevertheless demonstrably true. The Monty Hall problem is mathematically closely related to the earlier Three Prisoners problem and to the much older Bertrand's box paradox.

    When first presented with the Monty Hall problem an overwhelming majority of people assume that each door has an equal probability and conclude that switching does not matter (Mueser and Granberg, 1999). Out of 228 subjects in one study, only 13% chose to switch (Granberg and Brown, 1995:713). In her book The Power of Logical Thinking, vos Savant (1996, p. 15) quotes cognitive psychologist Massimo Piattelli-Palmarini as saying "... no other statistical puzzle comes so close to fooling all the people all the time" and "that even Nobel physicists systematically give the wrong answer, and that they insist on it, and they are ready to berate in print those who propose the right answer". Pigeons repeatedly exposed to the problem show that they rapidly learn always to switch, unlike humans (Herbranson and Schroeder, 2010).

    See also: The Three Prisoners Problem
    "The dogs bark but the caravan moves on."
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  6. #21
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    Quote Originally Posted by SoFlaLaw View Post
    sagefr0g,

    Thanks for the response. I guess that I will just agree to disagree. Your life or death example applies the 1/3 probability to the remaining non-selected door based upon a "values" (bad decision - do not reward the ploppy for playing the game in the first place with Mephisto, but reward the guy/gal for using their brain in trying to save their life now that it is at stake by applying the 1/3 probability to the non-selected door) judgment.

    To me, either way, the 1/3 probability from the exposed door can be applied to the initially selected door or the non-remaining door, thus I would assign 1/6 to each door (splitting that 1/3 probability), resulting in a 1:1 or 50% versus 50% probability. I do not care whether the initial decision was based on dumb decisionmaking or greed, or if the later decision was based upon a sincere effort to save a human soul (and win $1million to boot, or the devil's golden fiddle - ask Charlie Daniels Band if you do not understand this reference). Emotions and values judgments are irrelevant to my attempts to analyze this problem.

    Thanks for trying to shed light on my inability to grasp how this works according to most APs' explanations I have seen online since starting to CC.

    I appreciate the civil tone that all have thus far displayed regarding my inquiries.

    HNY
    lol, i figured (as i was driving to the joint) you'd bring up the 1/6 to each door (splitting that 1/3 probability), as well and i meant to incorporate that into my argument. forgot to, lol.
    i respect your lack of caring regarding emotions, value judgments and the like regarding the value assignable to a chances expectation, i suspect as well that you would value none the less a scientific approach regarding the assignment of said value. i don't think you'd argue with the fact that the first decision made could be nothing more than a gamble, which would in fact yield a value of a 1/3 chance for attaining the prize. such value would hold if the remaining two doors were checked simultaneously and regardless of the observational results, no?
    uhhhmm have you considered and read Aslan's excel 'simulation' of the problem? just curious what you would think of those results and his summation regarding same.
    best regards,
    mr fr0g MMOA honorary predator
    STRENGTH - HONOR - HEART
    that's my take on it your mileage may vary.
    for senior citizen fuzzy count click link:
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    Zengrifter,

    Thanks for your attempts to convince, but I guess I am amongst those who reject the commonly held "proof". Assuming that there were three (3) options, then each would have a 1/3 weight, and my initial selection would only have 1:2 (33 1/3%) odds, and the other 2 doors would hold 2:1 (66 2/3%) odds against my interests.

    When one of the 2 other options is removed, yes that changes the odds, but I guess I will never agree that if I am offered the option to switch that I would gain one-half (50%) of the 2:1 (66 2/3%) odds from the remaining option and the exposed option, thereby increasing my chances from my initial 1/3 to 2/3. The exposed option has been 100% proven to be a non-option (prize not behind door), so its removal must be accounted for, which means that there now are (and always were) only 2 options, 1 with the prize and 1 without the prize (1:1 or 50% odds).

    From my perspective, the 1/3 that was originally attached to the exposed option, cannot be provided solely to the remaining option, as that does not contemplate that it is equally possible that the prize is indeed behind my selected option (door). The only reason someone should attach the exposed option's 1/3 value to the remaining option, is if they know that the prize is behind the remaining option, in which case, my selected option will prove to be 0:1 (or 0:2 initially, either way 0%), and the remaining door would have 1:0 (100%) odds (no competition as the game show host knows that the remaining door has the prize). So assuming: (1) the contestant does not have that knowledge; AND (2) the contestant does not know that the game show host knows where the prize is; and (3) that the game show host DOES knows that the prize is behind the remaining door, THEN, it is my belief that the exposed option's 1/3 value has to be equally assigned to and split between my initially selected option, and the option available for me to switch. Assuming that any of these known "facts" (essential assumptions) are not truly known, then I am back to 1:1 (50% odds) with either my initial selection or the remaining switch option each being equally likely to contain the prize.

    I have studied this scenario and just cannot understand how those who advocate otherwise fail, refuse or have not internalized the need to take into account the changed fact of the exposed door, and the necessity of splitting its probability amongst the other 2 options (my initial selection and the remaining option to which I could switch), which is equivalent to removing it from the equation.

    Thanks for trying. If I am missing something about the 3 "facts" (necessary assumptions for the complete analysis), I would welcome your clear explanation as to what I have missed or am misunderstanding.

  8. #23

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    Quote Originally Posted by SoFlaLaw View Post
    If I am missing something about the 3 "facts" (necessary assumptions for the complete analysis), I would welcome your clear explanation as to what I have missed or am misunderstanding.
    What you are missing is the KNOWN: You KNOW that your initial selection is 1/3 likelihood...
    ... THAT does not change. How can 1/3 become 1/2?*

    *A veridical paradox produces a result that appears absurd but is demonstrated to be true nevertheless.
    Last edited by zengrifter; January 5th, 2015 at 11:18 PM.
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    Quote Originally Posted by zengrifter View Post
    What you are missing is the KNOWN: You KNOW that your initial selection is 1/3 likelihood...
    ... THAT does not change. How can 1/3 become 1/2?....
    i think how he makes 1/3 become 1/2 is he takes the 1/3 value away from the door that monty shows a goat behind thus eliminating it. anyway he takes that 1/3 splits it, or divides it in two then adds 1/6 to both remaining doors values. so 1/3 + 1/6 = 1/2 making 1/2 the new value for each remaining door.
    my argument is that the full 1/3 redeemed value should be added to the door left that we did not originally choose. reason being, we know we screwed up originally when we picked our first door, so it does not deserve to have any more value added to it. we are using facts that we know ie. information to make a determination of what should be done with the redeemed value. the fact that our first choice was unwise and taken from a mathematically disadvantageously large set needs to be recognized and taken into account for. this we do when we take the full 1/3 value and add it to the value of the unperturbed door.
    best regards,
    mr fr0g MMOA honorary predator
    STRENGTH - HONOR - HEART
    that's my take on it your mileage may vary.
    for senior citizen fuzzy count click link:
    http://www.youtube.com/watch?v=DrTiP4ZIUfI

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    Quote Originally Posted by sagefr0g View Post
    i think how he makes 1/3 become 1/2 is he takes the 1/3 value away from the door that monty shows a goat behind thus eliminating it. anyway he takes that 1/3 splits it, or divides it in two then adds 1/6 to both remaining doors values. so 1/3 + 1/6 = 1/2 making 1/2 the new value for each remaining door.
    my argument is that the full 1/3 redeemed value should be added to the door left that we did not originally choose. reason being, we know we screwed up originally when we picked our first door, so it does not deserve to have any more value added to it. we are using facts that we know ie. information to make a determination of what should be done with the redeemed value. the fact that our first choice was unwise and taken from a mathematically disadvantageously large set needs to be recognized and taken into account for. this we do when we take the full 1/3 value and add it to the value of the unperturbed door.
    sagefr0g,

    Your first paragraph correctly restated my approach to reallocating the value of the exposed door.

    As for your second paragraph, that is my point. People are making subjective (not objective) value based decisions on how to reallocate the value of the original probability of the exposed door (having a 1/3 probability before being exposed by the game show host). See my observations below regarding the subjectivity of your reasoning on this matter:

    1. "we know we screwed up originally when we picked our first door";
    2. "it does not deserve to have any more value added to it.";
    3. "we are using facts that we know...";
    4. "the fact that our first choice was unwise and taken from a mathematically disadvantageously large set needs to be recognized and taken into account for."

    As for 1 above, how? How do we know we "screwed up" when we made our initial selection? We don't and can't unless we can read the future (See, Nicholas Cage as Chris Johnson in Next) or the answer to the problem was disclosed to us (or we stole the information).

    As for 2 above, why? I can see where this would be true if our initial selection was made on a "known" bad choice. But since our initial selection was made in a truly random fashion, and not based on bad facts, it was done merely through gambling (cannot do more, but could choose not to play the game) on the odds 1:2 no matter which door we selected. So, how would our initial selection of Door 1, Door 2, or Door 3 merit "not deserv[ing] to have any more value added to it."?

    As for 3 above, what "facts" do we "know" (or more appropriately, what reasonable assumptions can be made)?

    Fact 1 - There are 3 doors;
    Fact 2 - Behind 1 door is a desired prize (new car, boat, cruise, etc.);
    Fact 3 - Behind the other 2 doors, there is nothing or a consolation prize (a cow, horse, goat, broken down old car, etc.);
    Fact 4 - We do not have any knowledge as to which door when exposed will contain the prize (AN ASSUMPTION);
    Fact 5 - The game show host has not told us that s/he knows behind which door the prize has been placed;
    Fact 6 - We do not know if the game show host "knows" behind which door the prize has been placed (if relevant at all);
    Fact 7 - We selected one door (call it Door 1), and not either of the other 2 doors (call them Door 2 and Door 3);
    Fact 8 - Our initial selection had a one out of three probability (1:2 odds or 33 1/3% chance) for success;
    Fact 9 - Each of the two doors that we did not select individually had the same probability for success;
    Fact 10 - Combined, the two doors that we did not select collectively had 2:1 odds or 66 2/3% chance for success;
    Fact 11 - We could not select BOTH of the other two doors, so we could not make a 2:1 odds or 66 2/3% selection;
    Fact 12 - The game show host reveals one of the other two doors (let's say that was Door 2);
    Fact 13 - The exposed door (Door 2) does not reveal the prize;
    Fact 14 - We are afforded the opportunity to stick with our initial selection (Door 1), or switch to the non-exposed door (Door 3);
    Fact 15 - The odds/probability of Door 2 (1:2 or 33 1/3%) for success no longer applies to Door 2 (proven not to contain the prize);
    Fact 16 - The nullification of Door 2's chance/probability for success can be taken into account;
    Fact 17 - Door 2's chance/probability for success can be nullified or reallocated;
    Fact 18 - If we nullify Door 2's chance/probability for success, Door 1 and Door 3 each have 1:1 odds or a 50% chance/probability for successfully selecting the prize;
    Fact 19 - If we split Door 2's chance/probability for success, Door 1 and Door 3 each receive 50% of Door 2's 1:2 odds or 33 1/3% chance/probability for successfully selecting the prize, resulting in each now having 1:1 odds or a 50% chance/probability for successfully selecting the prize; and
    Fact 20 - If we reallocate Door 2's chance/probability for success, the door (whether Door 1 or Door 3) to which it is reallocated has the advantageous 2:1 odds or a 66 2/3% chance probability for success, and the other door has the lesser 1:2 odds or a 33 1/3% chance probability for success.

    Someone here may generate additional "known" facts (assumptions). But I believe that I have identified most of the relevant assumptions.

    As for the fourth paragraph, how do we know our initial selection was "unwise"? How was that selection being taken from a "mathematically disadvantageously large set" different from whether we initially selected either of the other two doors? Or all you saying selecting any of the 3 doors initially was unwise? If so, then why would you reward any selection with Door 2's odds/probability for success?

    Even if you expand the set of options from 3 to 1000 doors, and expose 998 doors without showing the prize, each of those options initially were from (your words) a "mathematically disadvantageously large set" of options. But once the door(s) is(are) exposed, we have a relatively small set of options (Door 1 and Door 3, or if a 1000 Doors, Door 1 and Door 537), just 2.

    I understand your points, but disagree that you are arguing from a point of relevant facts. So, why does our "unwise" initial choice from a ""mathematically disadvantageously large set" deserve to be recognized and be taken into account (punished, definitely not an objective process)?

    I truly would like to understand your reasoning to attempt to internalize your position and see if I change my approach (an analytical process akin to selecting Door 1 versus Door 3, albeit there does not appear to have been a Door 2 in our discussion and analysis).
    Last edited by SoFlaLaw; January 6th, 2015 at 07:26 AM.

  11. #26
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    Default No Collusion, No Advantage

    SoFlaLaw must be an honest lawyer (if there can be such a thing) for not understanding the nature of collusion. The following article describes how the lack of collusion in the "Who Wants to Be a Millionaire" game results in a lack of advantage to switching.

    "Technically, the difference is that Monte Hall's probabilities are 'conditional,' while Regis Philbin's are absolute."

    http://krigman.casinocitytimes.com/a...l-problem-5601
    Last edited by Dodo; January 6th, 2015 at 12:34 PM.

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    Quote Originally Posted by Dodo View Post
    SoFlaLaw must be an honest lawyer (if there can be such a thing) for not understanding the nature of collusion. The following article describes how the lack of collusion in the "Who Wants to Be a Millionaire" game results in a lack of advantage to switching.

    "Technically, the difference is that Monte Hall's probabilities are 'conditional,' while Regis Philbin's are absolute."

    http://krigman.casinocitytimes.com/a...l-problem-5601
    a lawyer? really? naaawww (no offense SoFlaLaw just couldn't resist)
    interesting, but this stuff is making my head spin or is it just SoFlaLaw. (again no offense meant :-))
    i do get a kick out of the poem:
    What you think and what authenticate,
    Aren't always in compliance.
    That's why what you speculate,
    Runs second best to science.
    Last edited by sagefr0g; January 6th, 2015 at 01:42 PM. Reason: me make funny
    best regards,
    mr fr0g MMOA honorary predator
    STRENGTH - HONOR - HEART
    that's my take on it your mileage may vary.
    for senior citizen fuzzy count click link:
    http://www.youtube.com/watch?v=DrTiP4ZIUfI

  13. #28

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    Quote Originally Posted by sagefr0g View Post
    a lawyer? really? naaawww (no offense SoFlaLaw just couldn't resist)
    interesting, but this stuff is making my head spin or is it just SoFlaLaw. (again no offense meant :-))
    i do get a kick out of the poem:
    What you think and what authenticate,
    Aren't always in compliance.
    That's why what you speculate,
    Runs second best to science.
    Is this why pigeons perform better than humans?

    Why is the Monty Hall Dilemma so perplexing to humans, when mere pigeons seem to cope with it? Hebranson and Schroder think this is a case of our own vaunted intelligence working against us. When faced with a problem like this, we try to think it through, working out the best solution before we do anything. This would be fine, except we’re really quite bad at problems involving conditional probability (such as “if this happens, what are the odds of that happening?”). Despite our best attempts at reasoning, most of us arrive at the wrong answer.

    Pigeons, on the other hand, rely on experience to work out probabilities. They have a go, and they choose the strategy that seems to be paying off best. They also seem immune to a quirk of ours called “probability matching”. If the odds of winning by switching are two in three, we’ll switch on two out of three occasions, even though that’s a worse strategy than always switching. This is, of course, exactly what the students in Hebranson and Schroder’s experiments did. The pigeons, on the other hand, always switched – no probability matching for them.

    In short, pigeons succeed because they don’t over-think the problem. It’s telling that among humans, it’s the youngest students who do best at this puzzle. Eighth graders are actually more likely to work out the benefits of switching than older and supposedly wiser university students. Education, it seems, actually worsens our performance at the Monty Hall Dilemma.

    OR, is SoFlaLaw helping to cause us to see through some psuedo-math based MYTH?

    "The dogs bark but the caravan moves on."
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    The Zengrifter / James Grosjean Reputation Debate
    -----------------------------------------
    “Truth, like gold, is obtained not by growth, but by washing away all that is not gold.” — Leo Tolstoy........
    "Is everything a conspiracy? No, just the important stuff." ZG

  14. #29
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    Quote Originally Posted by SoFlaLaw View Post
    sagefr0g,

    Your first paragraph correctly restated my approach to reallocating the value of the exposed door.

    As for your second paragraph, that is my point. People are making subjective (not objective) value based decisions on how to reallocate the value of the original probability of the exposed door (having a 1/3 probability before being exposed by the game show host).
    well maybe, here is the thing though, people are at least doing something, at least really exerting some mental prowess in some regard over and on real phenomenon that exist, beyond just applying math with no justifiable reason.
    See my observations below regarding the subjectivity of your reasoning on this matter:

    1. "we know we screwed up originally when we picked our first door";
    2. "it does not deserve to have any more value added to it.";
    3. "we are using facts that we know...";
    4. "the fact that our first choice was unwise and taken from a mathematically disadvantageously large set needs to be recognized and taken into account for."

    As for 1 above, how? How do we know we "screwed up" when we made our initial selection? We don't and can't unless we can read the future (See, Nicholas Cage as Chris Johnson in Next) or the answer to the problem was disclosed to us (or we stole the information).
    just playing the game is a screw up. gambling is a screw up. mindlessly following monty’s orders is a screw up. ok, it’s not a screw up because it’s a free roll, right? well, let’s imagine the monty devil thing again, where behind two of the doors is DEATH. is it a screw up to play the game then? hypothesis proved: it’s a screw up.
    As for 2 above, why? I can see where this would be true if our initial selection was made on a "known" bad choice. there yah go!!! our initial selection was made on a “known” bad choice! But since our initial selection was made in a truly random fashion, and not based on bad facts, it was done merely through gambling (cannot do more, but could choose not to play the game) on the odds 1:2 no matter which door we selected. So, how would our initial selection of Door 1, Door 2, or Door 3 merit "not deserv[ing] to have any more value added to it."?
    merely gambling!? SoFlaLaw you didn’t just say that did you and you a card counter?
    As for 3 above, what "facts" do we "know" (or more appropriately, what reasonable assumptions can be made)?facts? reasonable assumptions? there yah go, now we are getting somewhere! and somewhere is where we want to get, not just spinning our wheels. hey it’s not a crystal clear perfect world out there in advantage play land but we do the best we can with what we’ve got, when we can. may seem like grasping at straws, but no out and out gambling, just gotta be thinking on your feet, savvy. i mean hey, you can go with your original losing gamble if you want, or you can use what information you have and refine your action, it’s up to you. science, my friend, but not rocket science.

    Fact 1 - There are 3 doors;
    Fact 2 - Behind 1 door is a desired prize (new car, boat, cruise, etc.); ect. = life & a million$
    Fact 3 - Behind the other 2 doors, there is nothing or a consolation prize (a cow, horse, goat, broken down old car, etc.); ect.= DEATH
    Fact 4 - We do not have any knowledge as to which door when exposed will contain the prize (AN ASSUMPTION);
    Fact 5 - The game show host has not told us that s/he knows behind which door the prize has been placed;
    Fact 6 - We do not know if the game show host "knows" behind which door the prize has been placed (if relevant at all); incorrect, we know & it’s relevant but I contend it is not necessary info as long as the prize is not exposed by monty. but we know, if nothing other than this problem is on the internet, lol. and if it’s on the internet the switch solution must be true. Bonjur? 
    Fact 7 - We selected one door (call it Door 1), and not either of the other 2 doors (call them Door 2 and Door 3);
    Fact 8 - Our initial selection had a one out of three probability (1:2 odds or 33 1/3% chance) for success;
    Fact 9 - Each of the two doors that we did not select individually had the same probability for success;
    Fact 10 - Combined, the two doors that we did not select collectively had 2:1 odds or 66 2/3% chance for success; YES!!!!
    Fact 11 - We could not select BOTH of the other two doors, so we could not make a 2:1 odds or 66 2/3% selection;
    Fact 12 - The game show host reveals one of the other two doors (let's say that was Door 2);
    Fact 13 - The exposed door (Door 2) does not reveal the prize; no it doesn’t, but it reveals the fact that door 3 now has a 2/3 value since door 2 has a zero value far as the prize goes
    Fact 14 - We are afforded the opportunity to stick with our initial selection (Door 1), or switch to the non-exposed door (Door 3);
    Fact 15 - The odds/probability of Door 2 (1:2 or 33 1/3%) for success no longer applies to Door 2 (proven not to contain the prize); no, we now know how much door 2 contributes to the 2/3 sum of door 2 & door 3, it contributes 0
    Fact 16 - The nullification of Door 2's chance/probability for success can be taken into account;
    Fact 17 - Door 2's chance/probability for success can be nullified or reallocated;
    Fact 18 - If we nullify Door 2's chance/probability for success, Door 1 and Door 3 each have 1:1 odds or a 50% chance/probability for successfully selecting the prize; NO, NO, NO! door 1 has a 1/3 chance when we chose it, door 2 and door 3 have a 2/3 chance together. just because door 2 ends up having a goat doesn’t mean all of a sudden that door2 and door 3 chances don’t equal 2/3.
    Fact 19 - If we split Door 2's chance/probability for success, Door 1 and Door 3 each receive 50% of Door 2's 1:2 odds or 33 1/3% chance/probability for successfully selecting the prize, resulting in each now having 1:1 odds or a 50% chance/probability for successfully selecting the prize; and there is no justifiable reason for us to do that math in fact we have information that makes our brains unable to justify adding any value to door 1
    Fact 20 - If we reallocate Door 2's chance/probability for success, the door (whether Door 1 or Door 3) to which it is reallocated has the advantageous 2:1 odds or a 66 2/3% chance probability for success, and the other door has the lesser 1:2 odds or a 33 1/3% chance probability for success.

    Someone here may generate additional "known" facts (assumptions). But I believe that I have identified most of the relevant assumptions.

    As for the fourth paragraph, how do we know our initial selection was "unwise"? How was that selection being taken from a "mathematically disadvantageously large set" different from whether we initially selected either of the other two doors? Or all you saying selecting any of the 3 doors initially was unwise? YES! or no, or maybe lol, but yes If so, then why would you reward any selection with Door 2's odds/probability for success? because one is either an advantage player to begin with and knows the score about monty, or one is trapped like a rat in a stupid move, but still is a smart enough rat to figure out how to maybe escape the trap. the trapped rat realizes his mistake and fixes it as best he can.

    Even if you expand the set of options from 3 to 1000 doors, and expose 998 doors without showing the prize, each of those options initially were from (your words) a "mathematically disadvantageously large set" of options. But once the door(s) is(are) exposed, we have a relatively small set of options (Door 1 and Door 3, or if a 1000 Doors, Door 1 and Door 537), just 2. true, the example just dramaticizes the error inherent in picking from a mathematically disadvantageously large set.

    I understand your points, but disagree that you are arguing from a point of relevant facts. So, why does our "unwise" initial choice from a ""mathematically disadvantageously large set" deserve to be recognized and be taken into account (punished, definitely not an objective process)? this is why i suggested the devil and his goat angels scenario. such a scenario proves the error of initially playing the game. that error may not be 100% fixable but one can improve one’s odds by using the information monty provided, and switching doors. but what? come on, have you never made a mistake, realized it was a mistake and went back and corrected it? that’s the idea of not putting any more value onto door 1, we realize it was a mistake in the first place. and on top of that we realize that there is no compelling reason to remove any value from the combined values of door 2 and door 3, just because door 2 ends up having zero value doesn’t change the combined value of door 2 and door 3.

    I truly would like to understand your reasoning to attempt to internalize your position and see if I change my approach (an analytical process akin to selecting Door 1 versus Door 3, albeit there does not appear to have been a Door 2 in our discussion and analysis).
    ok, tell yah what, i’ll switch, you are correct. 
    .........uhmm me in red, sorry, i'm can be kinda lazy
    Last edited by sagefr0g; January 6th, 2015 at 05:55 PM.
    best regards,
    mr fr0g MMOA honorary predator
    STRENGTH - HONOR - HEART
    that's my take on it your mileage may vary.
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  15. #30
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    As for your second paragraph, that is my point. People are making subjective (not objective) value based decisions on how to reallocate the value of the original probability of the exposed door (having a 1/3 probability before being exposed by the game show host).
    well maybe, here is the thing though, people are at least doing something, at least really exerting some mental prowess in some regard over and on real phenomenon that exist, beyond just applying math with no justifiable reason.
    Okkkkkay?


    As for 1 above, how? How do we know we "screwed up" when we made our initial selection? We don't and can't unless we can read the future (See, Nicholas Cage as Chris Johnson in Next) or the answer to the problem was disclosed to us (or we stole the information).
    just playing the game is a screw up. gambling is a screw up. mindlessly following monty’s orders is a screw up. ok, it’s not a screw up because it’s a free roll, right? well, let’s imagine the monty devil thing again, where behind two of the doors is DEATH. is it a screw up to play the game then? hypothesis proved: it’s a screw up.
    So what, continuing to play the game might be a screw up. How do you know the devil (if such a thing/entity really exists) isn't just Monty Hall playing more mind games with you, and that there is really no deadly option behind 1 of the 2 remaining doors, or perhaps death lurks behind both doors. Booyah!


    As for 2 above, why? I can see where this would be true if our initial selection was made on a "known" bad choice. there yah go!!! our initial selection was made on a “known” bad choice! But since our initial selection was made in a truly random fashion, and not based on bad facts, it was done merely through gambling (cannot do more, but could choose not to play the game) on the odds 1:2 no matter which door we selected. So, how would our initial selection of Door 1, Door 2, or Door 3 merit "not deserv[ing] to have any more value added to it."?
    merely gambling!? SoFlaLaw you didn’t just say that did you and you a card counter?
    Okay, I was sloppy. You got me there. Yes I am a CC. I can appreciate your characterization of the decision to play the game in the first place as being a bad choice, thereby rendering the selection of Door 1 as a bad decision, assuming that you are placing your life, health, loved ones or significant financial interests at risk. But that doesn't change the bad status had we selected either of the other 2 doors. ALL ARE BAD CHOICES, SINCE THEY ARE SELECTED WITHOUT AN ADVANTAGE AGAINST THE HOUSE.


    As for 3 above, what "facts" do we "know" (or more appropriately, what reasonable assumptions can be made)?facts? reasonable assumptions? there yah go, now we are getting somewhere! and somewhere is where we want to get, not just spinning our wheels. hey it’s not a crystal clear perfect world out there in advantage play land but we do the best we can with what we’ve got, when we can. may seem like grasping at straws, but no out and out gambling, just gotta be thinking on your feet, savvy. i mean hey, you can go with your original losing gamble if you want, or you can use what information you have and refine your action, it’s up to you. science, my friend, but not rocket science.
    Science and math is what I have been trying to get to the underlying fundamentals of, and what I have been utilizing (proving or disproving a hypothesis). I have no way of prejudging whether my initial selection was "your original losing gamble" or if Door 3 would be a "losing gamble".


    Fact 6 - We do not know if the game show host "knows" behind which door the prize has been placed (if relevant at all); incorrect, we know & it’s relevant but I contend it is not necessary info as long as the prize is not exposed by monty. but we know, if nothing other than this problem is on the internet, lol. and if it’s on the internet the switch solution must be true. Bonjur? 
    But he said he was a French model! Even if Monty Hall knows which door the prize is behind, provided that he exposes 1 of the 2 doors behind which the prize is not located, at best his knowledge is irrelevant. It could be behind the door I selected (Door 1) or the remaining door (Door 3 after he exposes Door 2).


    Fact 13 - The exposed door (Door 2) does not reveal the prize; no it doesn’t, but it reveals the fact that door 3 now has a 2/3 value since door 2 has a zero value far as the prize goes
    That is circular reasoning. Just because the prize is not behind Door 2 (now exposed), does not mean that its value automatically should be transferred to Door 3. That has been my point from the gitgo, and has not yet been disproven through objective method. Why can't its value be transferred to my initial selection of Door 1?

    Fact 15 - The odds/probability of Door 2 (1:2 or 33 1/3%) for success no longer applies to Door 2 (proven not to contain the prize); no, we now know how much door 2 contributes to the 2/3 sum of door 2 & door 3, it contributes 0
    Yes, it now contributes 0 to door 2, but likewise, before being exposed, it equally contributed to the 2/3 sum of Door 1 & Door 2.


    Fact 18 - If we nullify Door 2's chance/probability for success, Door 1 and Door 3 each have 1:1 odds or a 50% chance/probability for successfully selecting the prize; NO, NO, NO! door 1 has a 1/3 chance when we chose it, door 2 and door 3 have a 2/3 chance together. just because door 2 ends up having a goat doesn’t mean all of a sudden that door2 and door 3 chances don’t equal 2/3.
    You continue to refuse to acknowledge that although Door 2 & Door 3 had a 2/3 chance together, so did Door 1 & Door 2 OR Door 1 & Door 3. So, it means that there may be a goat (cow, sheep, lame horse, etc. or Monty Devil might kill you) behind either Door 1 or Door 3.


    Fact 19 - If we split Door 2's chance/probability for success, Door 1 and Door 3 each receive 50% of Door 2's 1:2 odds or 33 1/3% chance/probability for successfully selecting the prize, resulting in each now having 1:1 odds or a 50% chance/probability for successfully selecting the prize; and there is no justifiable reason for us to do that math in fact we have information that makes our brains unable to justify adding any value to door 1
    And you questioned my being a CC??? We always do the math! As for your conclusion, it is just that, a conclusion probably without any underlying solid foundation. The value from Door 2 can be added to either Door 1 or Door 3. You continue to refuse to recognize that possibility.


    As for the fourth paragraph, how do we know our initial selection was "unwise"? How was that selection being taken from a "mathematically disadvantageously large set" different from whether we initially selected either of the other two doors? Or all you saying selecting any of the 3 doors initially was unwise? YES! or no, or maybe lol, but yes If so, then why would you reward any selection with Door 2's odds/probability for success? because one is either an advantage player to begin with and knows the score about monty, or one is trapped like a rat in a stupid move, but still is a smart enough rat to figure out how to maybe escape the trap. the trapped rat realizes his mistake and fixes it as best he can.
    But the rat might have unknowingly fixed "it as best he can" when he made his initial selection of Door 1, and not switching to the possibly deadly decision awaiting behind Door 3. One thing you repeatedly disregarded in replying to my questions is that I assumed from the start that the game show host did not know the identity of the door behind which the prize (or life and $1 million) rested. Alternatively, I posited that it is irrelevant whether Monty knew where the prize rested, provided that he exposed 1 of the 2 doors behind which the prize was not located, and left my initial selection (Door 1) and the other door. Based upon that hypothetical that I posited, the rat cannot know behind which door the prize is located, and cannot know that switching would improve his odds (as his odds would grow to 1:1 or 50%) when the value of Door 2 is split between Door 1 and Door 3.


    Even if you expand the set of options from 3 to 1000 doors, and expose 998 doors without showing the prize, each of those options initially were from (your words) a "mathematically disadvantageously large set" of options. But once the door(s) is(are) exposed, we have a relatively small set of options (Door 1 and Door 3, or if a 1000 Doors, Door 1 and Door 537), just 2. true, the example just dramaticizes the error inherent in picking from a mathematically disadvantageously large set.
    I will leave this one alone, other than to state my objection to your characterization of the "mathematically disadvantageously large set" on the record based upon relevancy grounds.


    I understand your points, but disagree that you are arguing from a point of relevant facts. So, why does our "unwise" initial choice from a ""mathematically disadvantageously large set" deserve to be recognized and be taken into account (punished, definitely not an objective process)? this is why i suggested the devil and his goat angels scenario. such a scenario proves the error of initially playing the game. that error may not be 100% fixable but one can improve one’s odds by using the information monty provided, and switching doors. but what? come on, have you never made a mistake, realized it was a mistake and went back and corrected it? that’s the idea of not putting any more value onto door 1, we realize it was a mistake in the first place. and on top of that we realize that there is no compelling reason to remove any value from the combined values of door 2 and door 3, just because door 2 ends up having zero value doesn’t change the combined value of door 2 and door 3.
    Once again, you keep assuming that switching will fix the "mistake", but have failed to prove that the initial selection was indeed a mistake, or that it needed to be corrected. The assignment of Door 2's value solely to Door 3 is an assumption, without proof that it cannot be assigned to Door 1. Also, "just because door 2 ends up having zero value doesn't change the combined value of door 2 and door [1]". Come on, we're playing for my life!


    I truly would like to understand your reasoning to attempt to internalize your position and see if I change my approach (an analytical process akin to selecting Door 1 versus Door 3, albeit there does not appear to have been a Door 2 in our discussion and analysis).
    ok, tell yah what, i’ll switch, you are correct. 
    Au contraire. Maybe I'll switch, because your arguments appear on the internet! Any chance you are a french model?

    Thanks for trying, but I am still looking for proof or disproof (is that a legitimate word?) of my understanding of the Let's Make a Deal hypothetical.

    PS - Obviously, yes I am a lawyer, and I represent working people, their unions, pension plans, health and welfare plans and apprenticeship training programs. It's just the David in me that wants to defeat Goliath, just like when I take money away from the thieves, oops, meant casinos.

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