Toughest Game Show Question

Discussion in 'C Programming' started by divya bisht, Jan 10, 2012.

  1. divya bisht

    divya bisht Guest

    Toughest Game Show Question
    (Source : http://hardest-puzzle.blogspot.com/2012/01/toughest-game-show-question.html
    )

    You are on a game show and there are three doors. The presenter tells
    you that behind one of doors there is a car and behind the other two
    are goats, if you pick the car you win it. After you have picked a
    door the presenter opens a different door with a goat behind it, he
    then gives you the chance to change what door you open, what should
    you do?

    Discuss Solution in link below
    http://hardest-puzzle.blogspot.com/2012/01/toughest-game-show-question.html

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    divya bisht, Jan 10, 2012
    #1
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  2. divya bisht

    Bill Reid Guest

    On Jan 9, 8:56 pm, divya bisht <> wrote:
    >
    > You are on a game show and there are three doors. The presenter tells
    > you that behind one of doors there is a car and behind the other two
    > are goats, if you pick the car you win it. After you have picked a
    > door the presenter opens a different door with a goat behind it, he
    > then gives you the chance to change what door you open, what should
    > you do?
    >

    OH NO, THE DREADED "MONTY HALL" PROBLEM, IT'S BEEN AT LEAST A
    DECADE SINCE I'VE SEEN THIS MONSTROUSITY REAR IT'S UGLY HEAD IN
    USENET!!!!!!!

    The answer is that it doesn't matter, your odds are still one
    in three no matter what you do...but wait, does it matter if the
    presenter knows what's behind the doors and shows you the goat
    to make the game more interesting? Let's see what happens with
    the "regs" here, might be fun...

    ---
    William Ernest Reid
     
    Bill Reid, Jan 10, 2012
    #2
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  3. divya bisht

    tom st denis Guest

    On Jan 10, 9:47 am, Bill Reid <> wrote:
    > OH NO, THE DREADED "MONTY HALL" PROBLEM, IT'S BEEN AT LEAST A
    > DECADE SINCE I'VE SEEN THIS MONSTROUSITY REAR IT'S UGLY HEAD IN
    > USENET!!!!!!!
    >
    > The answer is that it doesn't matter, your odds are still one
    > in three no matter what you do...but wait, does it matter if the
    > presenter knows what's behind the doors and shows you the goat
    > to make the game more interesting?  Let's see what happens with
    > the "regs" here, might be fun...


    Never switch your bet. Pigeons know it.

    Tom
     
    tom st denis, Jan 10, 2012
    #3
  4. divya bisht

    Eric Sosman Guest

    On 1/10/2012 9:47 AM, Bill Reid wrote:
    > On Jan 9, 8:56 pm, divya bisht<> wrote:
    >>
    >> You are on a game show and there are three doors. The presenter tells
    >> you that behind one of doors there is a car and behind the other two
    >> are goats, if you pick the car you win it. After you have picked a
    >> door the presenter opens a different door with a goat behind it, he
    >> then gives you the chance to change what door you open, what should
    >> you do?
    >>

    > OH NO, THE DREADED "MONTY HALL" PROBLEM, IT'S BEEN AT LEAST A
    > DECADE SINCE I'VE SEEN THIS MONSTROUSITY REAR IT'S UGLY HEAD IN
    > USENET!!!!!!!
    >
    > The answer is that it doesn't matter, your odds are still one
    > in three no matter what you do...but wait, does it matter if the
    > presenter knows what's behind the doors and shows you the goat
    > to make the game more interesting? Let's see what happens with
    > the "regs" here, might be fun...


    Stick with the original choice. You can't eat a car.

    --
    Eric Sosman
    d
     
    Eric Sosman, Jan 10, 2012
    #4
  5. divya bisht

    John Gordon Guest

    In <> Bill Reid <> writes:

    > The answer is that it doesn't matter, your odds are still one
    > in three no matter what you do


    It does matter. If you switch, you have a two-in-three chance of winning.

    > ..but wait, does it matter if the presenter knows what's behind the
    > doors and shows you the goat to make the game more interesting?


    The presenter does know. He doesn't open a random door.

    --
    John Gordon A is for Amy, who fell down the stairs
    B is for Basil, assaulted by bears
    -- Edward Gorey, "The Gashlycrumb Tinies"
     
    John Gordon, Jan 10, 2012
    #5
  6. On 10-Jan-12 08:47, Bill Reid wrote:
    > On Jan 9, 8:56 pm, divya bisht <> wrote:
    >> You are on a game show and there are three doors. The presenter tells
    >> you that behind one of doors there is a car and behind the other two
    >> are goats, if you pick the car you win it. After you have picked a
    >> door the presenter opens a different door with a goat behind it, he
    >> then gives you the chance to change what door you open, what should
    >> you do?
    >>

    > OH NO, THE DREADED "MONTY HALL" PROBLEM, IT'S BEEN AT LEAST A
    > DECADE SINCE I'VE SEEN THIS MONSTROUSITY REAR IT'S UGLY HEAD IN
    > USENET!!!!!!!
    >
    > The answer is that it doesn't matter, your odds are still one
    > in three no matter what you do...but wait, does it matter if the
    > presenter knows what's behind the doors and shows you the goat
    > to make the game more interesting? Let's see what happens with
    > the "regs" here, might be fun...


    Mythbusters recently tested this. You're always better off switching.

    S


    --
    Stephen Sprunk "God does not play dice." --Albert Einstein
    CCIE #3723 "God is an inveterate gambler, and He throws the
    K5SSS dice at every possible opportunity." --Stephen Hawking
     
    Stephen Sprunk, Jan 10, 2012
    #6
  7. divya bisht

    Seebs Guest

    On 2012-01-10, John Gordon <> wrote:
    > The presenter does know. He doesn't open a random door.


    Our spammy friend seems to have a religious objection to an unambiguous
    problem statement, and has not specified this.

    -s
    --
    Copyright 2011, all wrongs reversed. Peter Seebach /
    http://www.seebs.net/log/ <-- lawsuits, religion, and funny pictures
    http://en.wikipedia.org/wiki/Fair_Game_(Scientology) <-- get educated!
    I am not speaking for my employer, although they do rent some of my opinions.
     
    Seebs, Jan 10, 2012
    #7
  8. divya bisht

    Shao Miller Guest

    On 1/10/2012 09:47, Bill Reid wrote:
    > On Jan 9, 8:56 pm, divya bisht<> wrote:
    >>
    >> You are on a game show and there are three doors. The presenter tells
    >> you that behind one of doors there is a car and behind the other two
    >> are goats, if you pick the car you win it. After you have picked a
    >> door the presenter opens a different door with a goat behind it, he
    >> then gives you the chance to change what door you open, what should
    >> you do?
    >>

    > OH NO, THE DREADED "MONTY HALL" PROBLEM, IT'S BEEN AT LEAST A
    > DECADE SINCE I'VE SEEN THIS MONSTROUSITY REAR IT'S UGLY HEAD IN
    > USENET!!!!!!!
    >


    One can download the following book "Information Theory, Inference, and
    Learning Algorithms" by David J. C. MacKay from his site:

    http://www.inference.phy.cam.ac.uk/mackay/itila/

    Or better yet, one can purchase it and have it shipped.

    This problem is discussed on page 57, exercise 3.8, if I'm not mistaken.

    However, the above-stated details are a little different.

    - Shao Miller
     
    Shao Miller, Jan 10, 2012
    #8
  9. Seebs <> writes:
    > On 2012-01-10, John Gordon <> wrote:
    >> The presenter does know. He doesn't open a random door.

    >
    > Our spammy friend seems to have a religious objection to an unambiguous
    > problem statement, and has not specified this.


    The question does say that the presenter opens a door with a goat behind
    it.

    --
    Keith Thompson (The_Other_Keith) <http://www.ghoti.net/~kst>
    Will write code for food.
    "We must do something. This is something. Therefore, we must do this."
    -- Antony Jay and Jonathan Lynn, "Yes Minister"
     
    Keith Thompson, Jan 10, 2012
    #9
  10. divya bisht

    James Kuyper Guest

    On 01/10/2012 04:27 PM, Keith Thompson wrote:
    > Seebs <> writes:
    >> On 2012-01-10, John Gordon <> wrote:
    >>> The presenter does know. He doesn't open a random door.

    >>
    >> Our spammy friend seems to have a religious objection to an unambiguous
    >> problem statement, and has not specified this.

    >
    > The question does say that the presenter opens a door with a goat behind
    > it.


    Yes, but it doesn't say whether he opened a random door that happened to
    have a goat behind it, or whether he deliberately choose a door that he
    knew had a goat behind it. The latter is, I believe, the intended
    situation in the standard "Monty Hall" problem. A lot of very confusing
    discussions about that problem have been based upon choosing a different
    assumption. To take the most extreme case: what if the presenter only
    bothers opening the second door if you happen to have already chosen the
    correct door, and otherwise leaves it closed (obviously, you've not been
    informed of this policy)?
     
    James Kuyper, Jan 10, 2012
    #10
  11. divya bisht

    Seebs Guest

    On 2012-01-10, Keith Thompson <> wrote:
    > Seebs <> writes:
    >> On 2012-01-10, John Gordon <> wrote:
    >>> The presenter does know. He doesn't open a random door.


    >> Our spammy friend seems to have a religious objection to an unambiguous
    >> problem statement, and has not specified this.


    > The question does say that the presenter opens a door with a goat behind
    > it.


    But it does not say whether we know that the presenter ALWAYS does this,
    or whether it's just that, on this single case, the presenter HAPPENED to
    do this.

    Imagine that you were running a game show for people who are familiar with
    this problem; the host strategy would be to always show people a goat if they
    picked the car, and not show them a goat if they already picked one. Then
    people who know the problem in the formulation where the host *must* always
    show you a goat will end up losing every time, because they only get the option
    to switch when doing so will hurt them.

    In the real show, the host chose whether or not to reveal another door in
    a way that made it not obvious that you were better off switching, because
    part of it was his read of the contestants.

    -s
    --
    Copyright 2011, all wrongs reversed. Peter Seebach /
    http://www.seebs.net/log/ <-- lawsuits, religion, and funny pictures
    http://en.wikipedia.org/wiki/Fair_Game_(Scientology) <-- get educated!
    I am not speaking for my employer, although they do rent some of my opinions.
     
    Seebs, Jan 10, 2012
    #11
  12. divya bisht

    Bill Reid Guest

    On Jan 10, 9:21 am, Stephen Sprunk <> wrote:
    > On 10-Jan-12 08:47, Bill Reid wrote:
    > > On Jan 9, 8:56 pm, divya bisht <> wrote:

    >
    > >> You are on a game show and there are three doors. The presenter tells
    > >> you that behind one of doors there is a car and behind the other two
    > >> are goats, if you pick the car you win it. After you have picked a
    > >> door the presenter opens a different door with a goat behind it, he
    > >> then gives you the chance to change what door you open, what should
    > >> you do?

    >
    > > OH NO, THE DREADED "MONTY HALL" PROBLEM, IT'S BEEN AT LEAST A
    > > DECADE SINCE I'VE SEEN THIS MONSTROUSITY REAR IT'S UGLY HEAD IN
    > > USENET!!!!!!!

    >
    > > The answer is that it doesn't matter, your odds are still one
    > > in three no matter what you do...but wait, does it matter if the
    > > presenter knows what's behind the doors and shows you the goat
    > > to make the game more interesting?  Let's see what happens with
    > > the "regs" here, might be fun...

    >
    > Mythbusters recently tested this.  You're always better off switching.
    >

    Did they use a Monte Carlo simulation where they fired cannonballs
    through the garage doors of dozens of innocent people to see whether
    there was a goat or car in the garage?

    You're always better off switching, EXCEPT if the presenter
    knows you've chosen the car and ONLY offers you a different
    door to potentially keep you from winning the car. In THAT case,
    you WILL win the car 1/3 of the time overall, and will win
    100% of the time when you decline the switch...so the behavior
    of the presenter DOES make a difference...

    ---
    William Ernest Reid
     
    Bill Reid, Jan 10, 2012
    #12
  13. In article <>,
    tom st denis <> wrote:
    >On Jan 10, 9:47 am, Bill Reid <> wrote:
    >> OH NO, THE DREADED "MONTY HALL" PROBLEM ...


    Ooooh, I love that problem. I've made money on that one.

    >Never switch your bet. Pigeons know it.


    That's why they're called "pigeons".

    The key is that Monty knows where the goats are, and thus
    introduces information into the game when he opens a door.
    Thus, you should always switch. Math on request.

    --
    -Ed Falk,
    http://thespamdiaries.blogspot.com/
     
    Edward A. Falk, Jan 11, 2012
    #13
  14. divya bisht

    Kaz Kylheku Guest

    On 2012-01-10, Bill Reid <> wrote:
    > On Jan 9, 8:56 pm, divya bisht <> wrote:
    >>
    >> You are on a game show and there are three doors. The presenter tells
    >> you that behind one of doors there is a car and behind the other two
    >> are goats, if you pick the car you win it. After you have picked a
    >> door the presenter opens a different door with a goat behind it, he
    >> then gives you the chance to change what door you open, what should
    >> you do?
    >>

    > OH NO, THE DREADED "MONTY HALL" PROBLEM, IT'S BEEN AT LEAST A
    > DECADE SINCE I'VE SEEN THIS MONSTROUSITY REAR IT'S UGLY HEAD IN
    > USENET!!!!!!!
    >
    > The answer is that it doesn't matter, your odds are still one
    > in three no matter what you do...but wait, does it matter if the
    > presenter knows what's behind the doors and shows you the goat
    > to make the game more interesting? Let's see what happens with
    > the "regs" here, might be fun...


    I think you an easily solve this by a simple division into cases,
    and simple probabilities.

    Look Ma, no conditional probability, no Bayes' theorem.


    Suppose Monty always reveals the goat. (This builds suspense in the
    audience and is good for the show's ratings, which are more important
    than whether or not cars are given away.)


    There are two cases:

    Case 1: (p = 2/3) You picked the goat.

    Monty knows that this case is occurring, and reveals to you the other goat! So
    you must switch to get the car, and by switching the car is guaranteed.

    Case 2: (p = 1/3) You picked the car.

    In this case, you must not switch; you're already on the car.
    Of course Monty reveals a goat in this case also.


    Now you don't know which of these cases you are in. But you do know
    that one of these cases requires switching and the other requires
    staying and that the one that requires switching occurs with p = 2/3, and the
    one that requires staying occurs with p = 1/3.

    This essentially translates to "the probability that you must switch in order
    to win the car is 2/3", and that translates to "the probability that
    the car is behind the door you did not pick in round 1 is 2/3".

    If you do not switch, you are betting on being in Case 2, which is p = 1/3: the
    same odds as a one-round draw.

    If you switch, you are betting that you are in Case 1, where you have p = 2/3
    odds, an improvement.



    Switching: no brainer.
     
    Kaz Kylheku, Jan 11, 2012
    #14
  15. On 10-Jan-12 17:43, Bill Reid wrote:
    > On Jan 10, 9:21 am, Stephen Sprunk <> wrote:
    >> On 10-Jan-12 08:47, Bill Reid wrote:
    >>> OH NO, THE DREADED "MONTY HALL" PROBLEM, IT'S BEEN AT LEAST A
    >>> DECADE SINCE I'VE SEEN THIS MONSTROUSITY REAR IT'S UGLY HEAD IN
    >>> USENET!!!!!!!
    >>>
    >>> The answer is that it doesn't matter, your odds are still one
    >>> in three no matter what you do...but wait, does it matter if the
    >>> presenter knows what's behind the doors and shows you the goat
    >>> to make the game more interesting? Let's see what happens with
    >>> the "regs" here, might be fun...

    >>
    >> Mythbusters recently tested this. You're always better off switching.
    >>

    > Did they use a Monte Carlo simulation where they fired cannonballs
    > through the garage doors of dozens of innocent people to see whether
    > there was a goat or car in the garage?


    They did a Monte Carlo simulation, but unfortunately nothing was
    destroyed in the process.

    I prefer the mathematical proof, but that doesn't make for good TV.

    > You're always better off switching, EXCEPT if the presenter knows
    > you've chosen the car and ONLY offers you a different door to
    > potentially keep you from winning the car.


    Their test scenario was that the presenter _always_ opens a losing door,
    which they said is how the actual game show worked.

    They also first tested how often people switched after that door was
    opened; interestingly, zero (of twenty) contestants did so.

    S


    --
    Stephen Sprunk "God does not play dice." --Albert Einstein
    CCIE #3723 "God is an inveterate gambler, and He throws the
    K5SSS dice at every possible opportunity." --Stephen Hawking
     
    Stephen Sprunk, Jan 11, 2012
    #15
  16. divya bisht

    Seebs Guest

    On 2012-01-11, Stephen Sprunk <> wrote:
    > Their test scenario was that the presenter _always_ opens a losing door,
    > which they said is how the actual game show worked.


    They are wrong, I'm pretty sure. One of the discussions of this included
    an interview with the guy, and there was no such rule.

    -s
    --
    Copyright 2011, all wrongs reversed. Peter Seebach /
    http://www.seebs.net/log/ <-- lawsuits, religion, and funny pictures
    http://en.wikipedia.org/wiki/Fair_Game_(Scientology) <-- get educated!
    I am not speaking for my employer, although they do rent some of my opinions.
     
    Seebs, Jan 11, 2012
    #16
  17. divya bisht

    Kaz Kylheku Guest

    On 2012-01-11, Seebs <> wrote:
    > On 2012-01-11, Stephen Sprunk <> wrote:
    >> Their test scenario was that the presenter _always_ opens a losing door,
    >> which they said is how the actual game show worked.

    >
    > They are wrong, I'm pretty sure. One of the discussions of this included
    > an interview with the guy, and there was no such rule.


    But odds are 2/3 that he lied in that interview.
     
    Kaz Kylheku, Jan 11, 2012
    #17
  18. divya bisht

    Dann Corbit Guest

    In article <>, says...
    [snip]
    Yeah, but what it you happen to *want* the goat?
     
    Dann Corbit, Jan 11, 2012
    #18
  19. divya bisht

    James Kuyper Guest

    On 01/11/2012 12:00 AM, Seebs wrote:
    > On 2012-01-11, Stephen Sprunk <> wrote:
    >> Their test scenario was that the presenter _always_ opens a losing door,
    >> which they said is how the actual game show worked.

    >
    > They are wrong, I'm pretty sure. One of the discussions of this included
    > an interview with the guy, and there was no such rule.
    >
    > -s

    <http://en.wikipedia.org/wiki/Monty_Hall_problem> contains a section
    titled "extended problem description" which fills out the unstated
    assumptions behind the original statement of the problem. There's a fair
    number of them.
    There's also a section titled "other host behaviors" which summarizes
    the results when other assumptions are made.
    --
    James Kuyper
     
    James Kuyper, Jan 11, 2012
    #19
  20. divya bisht

    James Kuyper Guest

    On 01/11/2012 02:48 AM, Dann Corbit wrote:
    > In article <>, says...
    > [snip]
    > Yeah, but what it you happen to *want* the goat?


    That's easy - switch to the door opened by the Host.
    --
    James Kuyper
     
    James Kuyper, Jan 11, 2012
    #20
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