The 3 Door Problem

Discussion in 'PHYSICS & MATHEMATICS' started by Terry M, Dec 2, 2012.

  1. Terry M

    Terry M Member Mushroom Doctor

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    This is a classic probability problem which has caused many a math professor to tear out her hair.

    You are on the show “Let’s Make A Deal.†You are given a choice of 3 doors. Behind one door is a brand new car!

    You choose door #1. The host shows you there is nothing behind door #2. Given the current probability, do you switch doors or not? Why?
     
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  2. Ozzy

    Ozzy Ggt's fanboi Moderator Mushroom Doctor Cannabis Doctor

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    hrmm..it sounds like I am oversimplifying this, but it seems as though there is a 50/50 chance of getting the car behind the door you pick. I am gonna go with my gut and stay with door number one. did I win my new mustang?
     
  3. Jeff

    Jeff Well-Known Member Moderator Mushroom Doctor Supporter

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    The Monty Hall problem. :thumbup:
     
  4. rogue

    rogue ♥ Hooked on Mycelium ♥ Moderator Mushroom Doctor Supporter

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    I'll agree with ozzy. Did he win?
     
  5. Jeff

    Jeff Well-Known Member Moderator Mushroom Doctor Supporter

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    If he stays where he is he has a 1 in 3 chance of winning.
     
  6. rogue

    rogue ♥ Hooked on Mycelium ♥ Moderator Mushroom Doctor Supporter

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    there was a one in three chance when it started but now that one has been revealed there is a 50:50 chance. No?
     
  7. Jeff

    Jeff Well-Known Member Moderator Mushroom Doctor Supporter

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    Not if he stays. He has the same odds even though one was taken away. He does have a 2 in 3 chance of winning if he switches though.
     
  8. Ozzy

    Ozzy Ggt's fanboi Moderator Mushroom Doctor Cannabis Doctor

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    yeah, I googled it up, I opted to answer before hand though in all fairness, as I will try to do with all questions posted in this forum, otherwise It wouldn't be any fun. Still, It stands to reason that If he knows that 2 is eliminated, that still leaves a probability of 50/50. I am failing to see how choosing to between 2 possibilities gives you you a 2/3 chance,even after reading through the correct answer.. guess that's why math was never my strong suit..lol
     
  9. rogue

    rogue ♥ Hooked on Mycelium ♥ Moderator Mushroom Doctor Supporter

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  10. Jeff

    Jeff Well-Known Member Moderator Mushroom Doctor Supporter

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    We had this problem in statistics in college. I knew the answer but I couldn't remember why. I had to cheat a little. :shrug1:
     
  11. Ozzy

    Ozzy Ggt's fanboi Moderator Mushroom Doctor Cannabis Doctor

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    I want to murder your math loving face jeff...
     
  12. Jeff

    Jeff Well-Known Member Moderator Mushroom Doctor Supporter

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    I have a BB gun and I am not afraid to use it.
     
  13. Ozzy

    Ozzy Ggt's fanboi Moderator Mushroom Doctor Cannabis Doctor

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    oh, really...


    *lowers head and crosses to other side of the street*
     
  14. Jeff

    Jeff Well-Known Member Moderator Mushroom Doctor Supporter

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    How did it come to this?
    6 months ago you wanted to have my love child.
     
  15. Ozzy

    Ozzy Ggt's fanboi Moderator Mushroom Doctor Cannabis Doctor

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    It's the crank man. It makes me lash out
     
  16. Terry M

    Terry M Member Mushroom Doctor

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    It's really hard to explain the solution so it makes intuitive sense. I realized that as soon as I posted it. Here's a way I thought of earlier today.

    Suppose that instead of 3 doors there are 100 door. You choose one, because the odds are 1 out of 100 for each and every door.

    Now the game show host, who knows what door the car is behind, opens 98 of the other 99 doors for you, showing that there's no car behind any of them. There is just your chosen closed door and the remaining closed door.

    Are the odds now 50/50? Or did the fact that the guy opened 98 doors that he knew the car wasn't behind give you some new information?

    It's the same thing with 3 doors as it is for 100.
     
  17. Jeff

    Jeff Well-Known Member Moderator Mushroom Doctor Supporter

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    It is interesting and you are right it is not intuitive. When you put it in the 100 door scenerio it is easier to grasp though. You start off with a 1 in 100 chance. He eliminates 98 variables. You now have a 98 in 100 chance of winning if you switch doors.

    Edit: whoops, that would be 99 in 100.
     
  18. rogue

    rogue ♥ Hooked on Mycelium ♥ Moderator Mushroom Doctor Supporter

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    i use to like math :(
     
  19. Terry M

    Terry M Member Mushroom Doctor

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    Yup.

    I remember when this problem appeared in a national newspaper column. The columnist gave the right answer. But a bunch of university math professors wrote to her, insisting that she was wrong and the odds had to be 50-50!

    I think there's a purely math way of proving this involving mutual information and conditional probabilities. But I don't know the details, and of course it's not going to provide much of any insight into the "why." Though it might have convinced some of the math professors. :)
     
  20. Jeff

    Jeff Well-Known Member Moderator Mushroom Doctor Supporter

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    Maybe this will help. Look at it conversely....if instead of taking one door away in the 3 door example they add 10 new doors. One of the original three has the car and the ten new ones are losers. Your odds are still 1 in 3 even though there are currently 13 doors. Your odds never change from the original guess. Same thing applies if you take doors away.