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Thread: Mythbusters- Monty Hall 3 Door Pick

  1. #61
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    Quote Originally Posted by sagefr0g View Post
    thank you Dodo. so if i'm understanding Benford's law at all, it appears the first choice will tend to be the smallest number so one should switch, in the two hand choice game described in Quanta, no?
    No, you would need to look at the digits of the "chosen" number. A model based on something like Benford's law (as a function of the digits of the seen number) would produce a likely distribution of the space of all possible numbers that include the seen and unseen ones. Then the answer become trivial. Keep the seen number if it is larger than the (weighted) average number in the model space; otherwise, choose the unseen one.

    I leave the creation of a model distribution space as a function of the seen number as an exercise for you. (I.e., it is beyond my ability to do this step.)

  2. #62
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    Cool Roll Your Own

    Quote Originally Posted by Dodo View Post
    Apparently, the Esquire thinks the programmed game simulators cheat if my ability to interpret legalese is accurate...
    Mgr. Aslan describes a spreadsheet that he made in posts #7 and #8 of this thread. If ZG and Dr. D. make such a spreadsheet and convince the Esquire to make a wager using such a spreadsheet (whose construction can be easily described and verified to prevent cheating), please consider giving me a cut of the action, too. A spreadsheet can be set up like a multi-line video poker game with perhaps thirty lines. If switching creates 0 to 14 wins, the Esquire wins; with 15 to 30 wins, he loses. The Esquire would be foolish not to take the wager as he expects switching to create 10 wins on average.

  3. #63
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    Quote Originally Posted by Dodo View Post
    No, you would need to look at the digits of the "chosen" number. A model based on something like Benford's law (as a function of the digits of the seen number) would produce a likely distribution of the space of all possible numbers that include the seen and unseen ones. Then the answer become trivial. Keep the seen number if it is larger than the (weighted) average number in the model space; otherwise, choose the unseen one.

    I leave the creation of a model distribution space as a function of the seen number as an exercise for you. (I.e., it is beyond my ability to do this step.)
    yep, i knew i was a lost puppy far as Benford's law....................
    so a weighted average number in the model space, would that mean one would have to either observe someone playing the game so as to collect stats or actually play the game and collect stats while playing to come up with a weighted average? or maybe run a simulation, sorta thing?
    i tried my erroneous hold or switch strategy in excel, seemed as if it was just a 50/50 sorta thing far as it looked, using those strategies.Name:  switch.jpg
Views: 145
Size:  55.8 KBName:  held.jpg
Views: 141
Size:  54.6 KB
    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

  4. #64
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    Quote Originally Posted by sagefr0g View Post
    yep, i knew i was a lost puppy far as Benford's law....................
    so a weighted average number in the model space, would that mean one would have to either observe someone playing the game so as to collect stats or actually play the game and collect stats while playing to come up with a weighted average? or maybe run a simulation, sorta thing?
    No, you need to go up a level in abstraction. The "model" is not a simple one (in which every possible number would be equally likely); so, the "weighted average" takes this into account--think about integration in the calculus. And the model would be created uniquely for the seen number--this number is a clue as to the distribution of numbers in the model space that includes the seen and unseen numbers.

    Suppose you told me that you wrote down the size of your bankroll on a piece of paper and another number on a second piece of paper. If one of the pieces of paper is revealed to begin with a "9" digit, my guess would be that your bankroll size is written down on the other piece of paper. I reach this conclusion because Benford's law tells me that an arbitary decimal number has less than a 5% chance of starting with a "9" digit. If the seen number started with a "1" or a "2" digit, my guess would be that that number is the size of your bankroll by a similar reasoning process. This "guess which number is my bankroll size" game is much easier than the game described in the article. In that game, arbitrary numbers could be revealed, so no model can be created a priori. But upon seeing the first number, you can use it as a clue to the characteristics of the the unseen number and create a model space that reflects all the possibilities with appropriate weightings. I will emphasize that YOU can: this is beyond my abilities or desire to do so.

  5. #65
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    Quote Originally Posted by Dodo View Post
    No, you need to go up a level in abstraction. The "model" is not a simple one (in which every possible number would be equally likely); so, the "weighted average" takes this into account--think about integration in the calculus. And the model would be created uniquely for the seen number--this number is a clue as to the distribution of numbers in the model space that includes the seen and unseen numbers.

    Suppose you told me that you wrote down the size of your bankroll on a piece of paper and another number on a second piece of paper. If one of the pieces of paper is revealed to begin with a "9" digit, my guess would be that your bankroll size is written down on the other piece of paper. I reach this conclusion because Benford's law tells me that an arbitary decimal number has less than a 5% chance of starting with a "9" digit. If the seen number started with a "1" or a "2" digit, my guess would be that that number is the size of your bankroll by a similar reasoning process. This "guess which number is my bankroll size" game is much easier than the game described in the article. In that game, arbitrary numbers could be revealed, so no model can be created a priori. But upon seeing the first number, you can use it as a clue to the characteristics of the the unseen number and create a model space that reflects all the possibilities with appropriate weightings. I will emphasize that YOU can: this is beyond my abilities or desire to do so.
    i think that clarifies it a great deal for me Dodo, thank you. edit: interestingly a commenter of the article has mentioned Benford's law and discussed it's applicability
    ."Liam says:
    July 8, 2015 at 4:16 am

    You do not have “absolutely no information” – you have the first number. You can use this as a “guide” to the range of possible numbers. So if the first number is between 0 and 9, then you assume the other number is also less than 10, and guess accordingly – lower for numbers 6+, higher for 4 and below, random pick for 5.

    If the number shown has two digits, then you assume a larger range of possible numbers, and guess accordingly. If it is a high number in the range of digits given, pick lower. I.e if the number displayed is 89, you pick lower.

    In this, you are using an application of Benford’s law, about distribution of first numbers. This method will fail many times, in the first example, if you’re actually choosing either 0 or 1 and show the 1, say. But you’re looking for a method that gives you an edge over all the possible games played, and I would say this does.

    Things get rather more interesting if the first number revealed is zero. I suspect most people would then say “higher”, but if negative numbers are allowed, (as per an example above) then I’d say zero is the one time the first number revealed gives you no useful information.
    " end edit

    is it a sensible conclusion that if a casino game existed that had a feature about it that was comparably the same to that of the game in the article that it simply wouldn't be worth trying to exploit it for the typical advantage player?
    Last edited by sagefr0g; July 9th, 2015 at 12:16 AM.
    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

  6. #66
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    Question

    Quote Originally Posted by sagefr0g View Post
    is it a sensible conclusion that if a casino game existed that had a feature about it that was comparably the same to that of the game in the article that it simply wouldn't be worth trying to exploit it for the typical advantage player?
    Proposition bets at the sports book? Ms. Ttthreee might know.

  7. #67
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    Quote Originally Posted by Dodo View Post
    Proposition bets at the sports book? Ms. Ttthreee might know.
    right, but i guess the N0 for the game in the article or for any such game would maybe be astronomical, sorta thing, i guess?
    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

  8. #68
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    Default i can't answer it but

    i posted the following on the two hand two number game:
    sagef says:
    July 14, 2015 at 12:42 pm
    the information comes from randomness, just like everything else in the known universe. selection or observation collapses what we donít know and canít completely know eg. the randomness into information. seems there would be two ways to beat the game, one purely by the winning just happening eg. some really good luck, sort of like the forming of the universe and evolution and two perhaps gaining a mathematical advantage from information gained by selecting over time while employing logic to implement the advantage.

    @

    https://www.quantamagazine.org/20150...om-randomness/
    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

  9. #69
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    Default Monty Hall, Redux

    Quote Originally Posted by Dodo View Post
    Apparently, the Esquire thinks the programmed game simulators cheat if my ability to interpret legalese is accurate...
    Maybe the code at the following link cheats, too?

    http://code.activestate.com/recipes/...in=lang-python

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