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Math Help - Monty Hall goes Car-Goat-Key

  1. #1
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    Monty Hall goes Car-Goat-Key

    Ann and Art face three doors marked 1, 2, 3. Behind the doors are placed,
    randomly, a car, a goat, and a key. The couple wins the car if Art finds
    the car, then Ann finds the key.

    First, Art (with Ann removed from the scene) opens a door; if car is not
    behind it, he opens a second door. If he fails to find the car, they lose.

    If he does find the car, then all doors are closed and Ann gets to open a
    door; if key is not behind it, she is allowed to open a second door.
    If she fails to find the key, they lose.

    Ann and Art do not communicate except to make a plan beforehand.

    Using the best strategy, what is the probability that they win the car?

    Ok; I'll give you a hint: the probability (with best strategy) is 2/3.
    Now this puzzle becomes: HOW do they do it?
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  2. #2
    Senior Member MacstersUndead's Avatar
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    There are six cases in which the car, key and goat can be randomly organized.
    (C car, G goat, K key)

    C G K
    C K G
    K G C
    K C G
    G C K
    G K C

    The strategy could be they would use is that Art would choose Doors 1 and 2, while Ann will choose Doors 3 and 2. why?

    2/3 cases have C on the left or in the middle.
    2/3 cases have K on the right or in the middle

    1/2 cases have C on the left or in the middle AND K on the right or the middle... hmm.
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  3. #3
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    No Macsters, but a bit of the right idea.

    Remember that total probability is 2/3.
    Since it it 2/3 for Art to find car, then Ann must be able to always
    get Key at first door or 2nd door she opens.

    They need a plan that makes that possible.
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  4. #4
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    Bump de bump...

    No takers?
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  5. #5
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    Well, guess time's up!

    The plan:
    Bob picks 1:
    if it's a goat, he picks 2 next
    if it's a key, he picks 3 next

    Alice picks 3 ;
    if it's a goat, she picks 2 next
    if it's a car, she picks 1 next

    There are 6 combo's:
    C G K
    C K G
    G C K
    G K C
    K C G
    K G C

    These 2 combo's are impossible:
    K C G [1]
    G K C [2]
    These 2 combo's mean SUCCESS:
    C G K
    G C K
    Leaves 2 combo's:
    C K G [3]
    K G C [4]

    [3]: since [1] is impossible, then Alice knows
    if she gets G that C K G is only possibility.
    So she opens door2.

    [4]: since [2] is impossible, then Alice knows
    if she gets C that K G C is only possibility.
    So she opens door1.
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