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Thread: Let's pick a door..

  1. #21
    Master Bowman Bottom Dweller's Avatar
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    There are 2 chance events described. Each has a separate probability for the choices and the event situation has changed. The first choice has a 1 in 3 chance. The second has a 1 in 2 chance.
    Archery's a lot like Life .... a few see it as an opportunity to score cheap points; while the rest are focused on the ends that count.
    Genesis 21:20 And God was with the boy as he grew up in the wilderness. He became a skillful archer.

  2. #22
    digitus impudicus
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    This is why I love posting this sort of thing.

    This has been done to death since 1974 and people here think that they're coming up with new scenarios. Of course the host knows where the car is.
    This is a game show.

    I take it that nobody has actually bothered to do the proofs?
    Pretty embarrassing to go to all this trouble when the world has already worked out several times that it's not 50/50

    Do you guys even bother to read?

    How much proof does it actually take to absolutely demonstrate that it's not 50/50?

    Even the most basic level of reading or googling will present all the aspects of the Monty Hall problem in infinite detail and give basic instructions on how to carry it out.

    Even when I tell you that it's NOT intuitive, you continue to insist that it is. This is absolute proof that you've not bothered to actually test what you think is true because you'll very quickly find out that it isn't.

    You can even go to the link with the simulator and do it from the very desktop that you're reading this information on now.

    You could like, manually do the test ten times by sticking with your choice, or changing it.

    Right now, people reading through this page will be doing exactly this and discovering that in ten tries, two thirds of the time, they'll win the car by switching.

    They'll know absolutely that people saying that it's 50-50 are wrong, and so wrong that they're continuing to embarrass themselves by not doing the most basic level of investigation, even after being provided the tools to do so.

    This isn't a new thing. I'm not bringing out some special trick. This is a historical problem which always has the same result.

    The first video explains this very clearly why Monty is the key.






    Last edited by Andy!; 10th January 2017 at 07:03 AM.
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  3. #23
    Master Bowman Bottom Dweller's Avatar
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    Well there you go. What I failed to realise was that the second chance event is not a random chance event at all, but is influenced by the first event and the picking of the door by the host who importantly adds the non-random element because they know where the cars is and is not.

    Now if the 2 remaining doors had been randomly mixed again before offering the choice to change then it would be 50/50. But they have not been randomly mixed.
    Archery's a lot like Life .... a few see it as an opportunity to score cheap points; while the rest are focused on the ends that count.
    Genesis 21:20 And God was with the boy as he grew up in the wilderness. He became a skillful archer.

  4. #24
    Learning the Ropes
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    Nobody feel bad if they get this wrong. Professors of mathematics have got it wrong. I did when I first met it in 1985 or so.

    As Brenton says, the knowledge of the games show host is critical. If the door was opened at random and terminated if the prize was revealed, then with this fairly small change, there is no advantage to switching. This problem's noteriety arises in translating the English language description to a mathematical one. In this case, the description is not really ambiguous, but it invites the reader to jump to the wrong conclusion about what sort of problem it is. It also assumes some background knowledge about TV games, which might be more available to some than others!

    What I find interesting is the cognitive aspect of understanding the problem rather than the mathematics of solving it.

    Probability starts out trivial and gets tricky very fast!

  5. #25
    Learning the Ropes
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    Its still a 50/50 choice however if you want to win you should swap as sticking with your original choice only gives you a 33% chance as opposed to a 66% chance of winning (given that one of the 33% chances has been
    removed).

    As an asside I remember as a Kid(would be over 40 years ago) one of the main newspapers in Melbourne (SUN or Herald) ran a competition and all you had to do was pick between two Animals(I dont remember what they were) and if you picked the right amimal you won. Only one entry per household was allowed.

    No one won a prize in that competition.

    I assume some sort of social experiment was going on and the paper knew that no one or at least the majority would not pick the winning animal.

  6. #26
    Lowered expectations Stash's Avatar
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    Does it work with wives/husbands as well?

    Picked one a long time ago. Had a choice of 2 others, discovered that one of them would have been a very bad choice. Decided to stick with the first one. Jury still out if I made the right choice.

    OK, maybe not EXACTLY the same thing...

  7. #27
    Grand Master Bowman scott p's Avatar
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    Quote Originally Posted by craigr View Post
    Its still a 50/50 choice however if you want to win you should swap as sticking with your original choice only gives you a 33% chance as opposed to a 66% chance of winning (given that one of the 33% chances has been
    removed).
    that's the point I cannot see.

    I choose door 1.
    the host opens 2 door and shows it's not the prize.

    all I know is it's door 1 or 3. that's a 50% of winning. how does the elimination of door 2 in any way tell me that it's not door 1, and I have not picked correctly from the outset?

    it was only a 33% chance when I originally made the choice. it became a 50% choice when it became a choice between 2 doors.

  8. #28
    Slowly improving Brenton's Avatar
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    So when we assume my blue scenario above.. (the host KNOWS the car door and chooses to keep it in play by DELIBERATELY opening the sand/goat door)

    Then trick of this question that confuses people is the small sample of choices that goes from just 3 to 2.


    However it is really the same as this scenario....

    A bag contains 10 marbles, 1 black and 9 white ones....if you choose the black one you win a car.
    - The guest reaches in randomly and chooses a marble in his closed hand (can't see the marble yet) he has 1/10 chance of getting the black one.. will happen only 1 in 10 times
    - The host now LOOKS into the bag and 9/10 times he chooses the black marble and hides it in his hand (9/10 of the time he will have the black marble... 1/10 of the time he will have a white one because 1/10 times the black one is in the guests hand)

    Now which marble should the guest choose... the one in their hand... or the one in the host's hand. In this case the choice between selecting the marble in the guest's or host's hand is still 2 choices... but the probability the black marble being in the guest hand is is NOT 50/50.... it is still 1/10... you are far better off changing to the host marble because 9/10 time it will be the black one.

    The 3 door scenario is EXACTLY the same... with just lower (therefore more confusing) odds.
    Last edited by Brenton; 10th January 2017 at 11:53 AM.

  9. #29
    digitus impudicus
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    Quote Originally Posted by iandall View Post
    Nobody feel bad if they get this wrong. Professors of mathematics have got it wrong. I did when I first met it in 1985 or so. !
    Addressing famous problems like this is fascinating because you know that they're styled to be misunderstood before you get the answer which everyone gets wrong.

    Understanding how the process is intuitively going to lead you to the wrong conclusion is really cool.

    The thing that most people fail to grasp is that many issues are exactly the same as this, yet there is no existing structure to tell you that you're actually wrong in so simple a fashion.

    People draw conclusions from how six arrows are on the target or are convinced that some particular outcome definitely means something. This becomes conventional archery knowledge, yet everyone outside the archery world actually knows that this doesn't happen.

    As already demonstrated, even though this has been comprehensively proven, people are still clinging to the 50/50 split and WON'T understand it.

    Quote Originally Posted by iandall View Post
    What I find interesting is the cognitive aspect of understanding the problem rather than the mathematics of solving it.
    This is exactly why I love these sort of problems.
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  10. #30
    Grand Master Bowman scott p's Avatar
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    Something else... if I get a 66% chance by switching once one choice is eliminated, why don't I also have a 66% chance by retaining the initial choice?

    It's still one out of two no matter how it gets portrayed, and including the door the host opens one measure of chance but not the other is not a direct comparison.

  11. #31
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    Quote Originally Posted by scott p View Post
    But the host will always pick the door with the goat, so the choice is still 50:50.
    Yes your choice is 50/50 but your chances of winning are with changing.

    Look at it this way... when you make your first pick you have a 33% chance of winning with that pick and that never changes.

    The other 2 doors represent a 66% chance of winning(if you could choose both)

    When the host opens a door to reveal a goat your CHOICES are now 50/50 but your CHANCE of winning remains at 33% with your original pick and 66% with what is left(the other door).

    So the obvious thing to do is swap but thats still no guarantee of winning just a better CHANCE.

  12. #32
    digitus impudicus
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    Quote Originally Posted by craigr View Post
    Yes your choice is 50/50 but your chances of winning are with changing.

    Look at it this way... when you make your first pick you have a 33% chance of winning with that pick and that never changes.

    The other 2 doors represent a 66% chance of winning(if you could choose both)

    When the host opens a door to reveal a goat your CHOICES are now 50/50 but your CHANCE of winning remains at 33% with your original pick and 66% with what is left(the other door).

    So the obvious thing to do is swap but thats still no guarantee of winning just a better CHANCE.
    Yep. Nailed it.

    And if you do the simulation manually by always changing your mind, it doesn't take many tries to show you that you win the car twice as often as sticking with your first guess.
    The success rate of changing is obvious with less than ten tries.

    Looks like there's only one person left who can't understand that it's not 50/50..
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  13. #33
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    Quote Originally Posted by craigr View Post
    Yes your choice is 50/50 but your chances of winning are with changing.

    Look at it this way... when you make your first pick you have a 33% chance of winning with that pick and that never changes.

    The other 2 doors represent a 66% chance of winning(if you could choose both)

    When the host opens a door to reveal a goat your CHOICES are now 50/50 but your CHANCE of winning remains at 33% with your original pick and 66% with what is left(the other door).

    So the obvious thing to do is swap but thats still no guarantee of winning just a better CHANCE.
    for me this is where i feel the mistake is. because you now are not playing the same game (and numbers)as you were with 3 doors. you now only have a choice of two so the maths for the three doors can't apply. you have to start the maths again as only two choices apply, not three. its like saying that if you only had two doors to choose from you could pretend there was a third door there so the maths was in your favor.

  14. #34
    Grand Master Bowman scott p's Avatar
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    Quote Originally Posted by craigr View Post
    The other 2 doors represent a 66% chance of winning(if you could choose both)
    but I can't!

    When the host opens a door to reveal a goat your CHOICES are now 50/50 but your CHANCE of winning remains at 33% with your original pick and 66% with what is left(the other door).
    I always have to pick an initial door.
    I have zero idea if that is a winning door or not.
    The host will *always* open a losing door, regardless of whether I have chosen a winning door or not initially.
    So ultimately my choice comes down to one of two doors, and that's 50%.

    Quote Originally Posted by xracer View Post
    for me this is where i feel the mistake is. because you now are not playing the same game (and numbers)as you were with 3 doors. you now only have a choice of two so the maths for the three doors can't apply. you have to start the maths again as only two choices apply, not three. its like saying that if you only had two doors to choose from you could pretend there was a third door there so the maths was in your favor.
    exactly. who cares about the chances on any individual door open?

    all I care about is my overall chance of winning.

  15. #35
    Slowly improving Brenton's Avatar
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    Scott... you need to reread my marble example above... the choice still comes down to one of two marbles in the hands.... but the one in the guests hand is still NOT 50% chance of winning.

    The whole 50% or 66% depends TOTALLY on the host KNOWING where the prize is... and deliberately not opening the prize door... that fact changes the odds.

    If you still believe it's 50%... then I'll have a game of 3 cups for you to play next time we're at a tournament together... I'll let you keep your original choice every time and double your money every time you win.



    If you flip a coin to decide if you'll win lotto tonight (heads you'll win lotto, tails you won't win lotto)... on average you'll be right 50% of the time... BUT... your not going to win the lotto 50% of the time... you coin will be right virtually EVERY TIME it's tails.

    Poker machines... taxing people for being bad at maths for 130 years.

  16. #36
    Learning the Ropes
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    Quote Originally Posted by Brenton View Post
    Scott... you need to reread my marble example above... the choice still comes down to one of two marbles in the hands.... but the one in the guests hand is still NOT 50% chance of winning.

    The whole 50% or 66% depends TOTALLY on the host KNOWING where the prize is... and deliberately not opening the prize door... that fact changes the odds.

    If you still believe it's 50%... then I'll have a game of 3 cups for you to play next time we're at a tournament together... I'll let you keep your original choice every time and double your money every time you win.



    If you flip a coin to decide if you'll win lotto tonight (heads you'll win lotto, tails you won't win lotto)... on average you'll be right 50% of the time... BUT... your not going to win the lotto 50% of the time... you coin will be right virtually EVERY TIME it's tails.

    Poker machines... taxing people for being bad at maths for 130 years.
    I'm not sure switching to marbles really helps. Sticking with doors, imagine there are 100 doors. Then your initial choice has only a 1/100 chance of being correct. More to the point, it is almost certain (99/100) that the prize is behind one of the other 99 doors. The host now goes and opens 98 of the other doors which does nothing to make your original choice more likely. So now it is now 1/100 for the original door and 99/100 for the remaining other door.

    One way of thinking about this is the host knows where the prize is. By only opening non-prize doors, he is partially communicating his knowledge to you, which you can use to make a better than random decision.

    If, on the other hand, the host opens doors at random, then the original choice is still 1/100 (that never changes), the remaining choice is also 1/100 and there is a 98/100 chance the host reveals the prize (by "accident") terminating the competition. In this case, it wouldn't matter if you switched, but you would still be at the same old lousy 1/100 odds.

  17. #37
    digitus impudicus
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    I would like to put forward that if we have got to this point where there are significant examples of explanation including videos and multiple explanations of the same thing, that if someone doesn't believe that the odds have actually changed, that we just stop bothering to explain it.

    There are literally hundreds of explanations that demonstrate quite convincingly that the odds stay at 1/3 without switching and change to 2/3 with switching.

    You can explain things, but you can't understand things for other people.
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  18. #38
    Grand Master Bowman scott p's Avatar
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    Quote Originally Posted by iandall View Post
    I'm not sure switching to marbles really helps. Sticking with doors, imagine there are 100 doors. Then your initial choice has only a 1/100 chance of being correct. More to the point, it is almost certain (99/100) that the prize is behind one of the other 99 doors. The host now goes and opens 98 of the other doors which does nothing to make your original choice more likely. So now it is now 1/100 for the original door and 99/100 for the remaining other door.
    so we now have the original door, and the one remaining door.

    that's a 50/50 chance. all the removal of the known-incorrect doors has done is increase the chances of the initial choice being right.

  19. #39
    digitus impudicus
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    If you're still insisting that this is 50/50, despite the incredible amount of evidence proving otherwise, you're doing a very convincing job of proving you're thick.

    Please continue. You'll eventually sway everyone.
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  20. #40
    1st Class Matrix Makeover's Avatar
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    Quote Originally Posted by scott p View Post
    The host will *always* open a losing door, regardless of whether I have chosen a winning door or not initially.
    I think this is where you are coming unstuck. The host will know if you have picked the right door or not. To keep the suspense going, they will not open the door you first pick - so they have the option of two doors.

    IF you have picked the right door first, they can open either of the other two doors. So the host has a 50/50 choice.
    IF you have picked the wrong door first, then there is only one door that they can open (the other wrong door) because they again want to keep the suspense going. Thus the host has zero percent choice in which door they open.

    I will become unstuck with the maths, but the key is knowing the host isn't making a random choice themselves.

    Chris

    Edit - I think I glued the maths - because you have a 2/3rds chance of picking the wrong door first time, the host has to open the other wrong door 66% of the time?
    Last edited by Matrix Makeover; 11th January 2017 at 01:19 PM. Reason: had a brain wave.
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