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Thread: Monty Hall Paradox

  1. #1 Monty Hall Paradox 
    Time Lord zinjanthropos's Avatar
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    Monty Hall was the host of TV game show ‘Lets Make a Deal”. Contestants had to pick between 1 of 3 curtains to see what’s behind it…..well…. check out this Wiki piece before I mess up.

    https://en.m.wikipedia.org/wiki/Monty_Hall_problem

    Not sure if this particular subject has come up before. For us mathematically challenged, it’s a head scratcher but I think I get it after reading up on it. Marilyn vos Savant obviously had no problem with it. Problem has boggled the minds of many number enthusiasts and I’m not sure if the answer is accepted in some parts. Other factors come into play. Interesting if you haven’t seen it before.


    All that belongs to human understanding, in this deep ignorance and obscurity, is to be skeptical, or at least cautious; and not to admit of any hypothesis, whatsoever; much less, of any which is supported by no appearance of probability...Hume
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  3. #2  
    KJW
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    I don't know if it's been discussed here, but I have seen it discussed elsewhere. I'm quite familiar with the problem and do understand the solution. It has even been featured on Mythbusters.


    There are no paradoxes in relativity, just people's misunderstandings of it.
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  4. #3  
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    1+1=2 is a slightly easier version of the problem.
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    Time Lord zinjanthropos's Avatar
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    Quote Originally Posted by mathman View Post
    1+1=2 is a slightly easier version of the problem.
    Everyone loves simplicity Mathman.

    I think you’re saying having one of the 3 choices revealed doesn’t change the denominator. Revealed pick and original pick’s odds each remain at 1/3 but by switching, the new pick has 2/3 chance. Technically a 2nd pick thus the numerator becomes 1+1. Denominator remains at 3. That’s how I saw it but been wrong before.
    All that belongs to human understanding, in this deep ignorance and obscurity, is to be skeptical, or at least cautious; and not to admit of any hypothesis, whatsoever; much less, of any which is supported by no appearance of probability...Hume
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  6. #5  
    KJW
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    Instead of probability, consider a large number of trials, say 1000 trials:

    For all of the 1000 trials, I choose door A as my initial choice.

    Of these 1000 trials, the prize will be behind door A approximately 333 times, behind door B approximately 333 times, and behind door C approximately 333 times.

    Of the 333 times the prize is behind door A, door B will be revealed to be empty approximately 167 times, and door C will be revealed to be empty approximately 167 times.

    Of the 333 times the prize is behind door B, door C will be revealed to be empty all 333 times.

    Of the 333 times the prize is behind door C, door B will be revealed to be empty all 333 times.

    Thus, of the 1000 trials in total, door B will be revealed to be empty approximately 500 times, and door C will be revealed to be empty approximately 500 times.

    Of the 500 times door B was revealed to be empty, the prize was behind door A approximately 167 times, and not behind door A approximately 333 times.

    Of the 500 times door C was revealed to be empty, the prize was behind door A approximately 167 times, and not behind door A approximately 333 times.

    Thus, of the 1000 trials in total, the prize was behind door A approximately 333 times, and not behind door A approximately 667 times.

    Thus, revealing empty doors does not alter the fact that the prize is behind the initially chosen door only 1/3 of the time, and therefore behind the other door 2/3 of the time. What revealing empty doors does is reduce the number of doors other than the initially chosen door from two to one.
    There are no paradoxes in relativity, just people's misunderstandings of it.
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  7. #6  
    Time Lord zinjanthropos's Avatar
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    Wow! Gave that one a lot of thought KJ. Nice going.

    We did a fun experiment today. My friend put 3 cards face down on a table 100 times. Of the 3 cards , one was red suited. I would pick a card that might be red suited and he would then reveal one of the black suited cards leaving two still face down. I would switch my pick and reveal that card. By switching I was able to get the red card 62 times. We repeated experiment but I had to stick with my original pick and I was successful only 41 times. So it was fairly close to the 2/3 switch vs 1/3 stick success rate.
    All that belongs to human understanding, in this deep ignorance and obscurity, is to be skeptical, or at least cautious; and not to admit of any hypothesis, whatsoever; much less, of any which is supported by no appearance of probability...Hume
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  8. #7  
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    I recall reading about a version of this but with pigeons and food rewards. The pigeons figured it out quicker than most people.
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    Actually, this is not a paradox at all. It illustrates a basic point in probability theory, which is this: given that the probability of a event at time = t is p, nothing that subsequently happens - no later knowledge - changes p. Later events, given the new knowledge, may well have a different probability
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    Quote Originally Posted by Guitarist View Post
    Actually, this is not a paradox at all. It illustrates a basic point in probability theory, which is this: given that the probability of a event at time = t is p, nothing that subsequently happens - no later knowledge - changes p. Later events, given the new knowledge, may well have a different probability
    Actually Wiki calls it a paradox . From Wiki:

    According to Quine's classification of paradoxes:

    .
    All that belongs to human understanding, in this deep ignorance and obscurity, is to be skeptical, or at least cautious; and not to admit of any hypothesis, whatsoever; much less, of any which is supported by no appearance of probability...Hume
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  11. #10  
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    It's only a paradox if you don't understand probability...
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  12. #11  
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    Quote Originally Posted by PhDemon View Post
    It's only a paradox if you don't understand probability...
    Not arguing that. Reconfiguring Animal Farm...

    All selections have an equal chance but some selections have more of a chance than others..... Apologies to George Orwell fans. Not sure if it fits the definition of paradox. I think it is close
    All that belongs to human understanding, in this deep ignorance and obscurity, is to be skeptical, or at least cautious; and not to admit of any hypothesis, whatsoever; much less, of any which is supported by no appearance of probability...Hume
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    Quote Originally Posted by zinjanthropos View Post
    All selections have an equal chance but some selections have more of a chance than others...... Not sure if it fits the definition of paradox.
    I think it is close enough. But in mathematics and logic, if your thread of reasoning leads to a paradox, your reasoning is perforce wrong
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    There are obviously several ways of explaining this problem. The more intuitive to me is "trying" to pick a wrong door as first choice. I am 67% to succeed. The interesting thing about this problem is that it helps understand that probability do not exist in the the world in a situation like this, where everything is already in place when you start playing.If you choose randomly, your probablity at the second choice is 50%, (you are 50% likely to be 67% likely to win and 50% likely to be 33% likely to win, numbers work), but from the point of view of a competent probabilist observing you after you have chosen but before you open, you are either 67% to win or 33% likely to win. However, you are either 100% or 0% to win from the point a view of an omniscient observer. Average is always 50%! Does real probability exist in the world when we throw a coin? Whatever your answer to this question, the probability of you guessing the result of the coin throw will be 50% (unless you have some superpowers). So probabilty does not depend on the nature of the universe, but is a property of our representation of the world.
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  15. #14  
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    Quote Originally Posted by kiskrof View Post
    There are obviously several ways of explaining this problem. The more intuitive to me is "trying" to pick a wrong door as first choice. I am 67% to succeed. The interesting thing about this problem is that it helps understand that probability do not exist in the the world in a situation like this, where everything is already in place when you start playing.If you choose randomly, your probablity at the second choice is 50%, (you are 50% likely to be 67% likely to win and 50% likely to be 33% likely to win, numbers work), but from the point of view of a competent probabilist observing you after you have chosen but before you open, you are either 67% to win or 33% likely to win. However, you are either 100% or 0% to win from the point a view of an omniscient observer. Average is always 50%! Does real probability exist in the world when we throw a coin? Whatever your answer to this question, the probability of you guessing the result of the coin throw will be 50% (unless you have some superpowers). So probabilty does not depend on the nature of the universe, but is a property of our representation of the world.
    What is your point?
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