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Math Help - Explanation to a logical problem needed!

  1. #1
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    Explanation to a logical problem needed!

    There are 3 ants at 3 corners of a triangle, they randomly start moving towards another corner.. what is the probability that they don't collide.

    All three should move in the same direction - clockwise or anticlockwise. Probability is 1/4.

    Can someone please explain the answer?
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  2. #2
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    Label the triangle \Delta ABC. Any bit-triple will denote the direction the ant at the vertices goes. Example: \left( {0,1,1} \right) means that ant at A goes counter-clockwise while ants at B & C go clockwise. In that case ant A will collides with the ant at B. There are eight such triples. In how many will there be no collisions?
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  3. #3
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    Quote Originally Posted by janvdl View Post
    Maybe just misread there, Plato?
    No, I think that you misread it.
    That was an example in which they do collide.
    Did you read the last two sentences?
    There are eight such triples. In how many will there be no collisions?
    Now I ask you the same question.
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  4. #4
    Bar0n janvdl's Avatar
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    Quote Originally Posted by Plato View Post
    No, I think that you misread it.
    That was an example in which they do collide.
    My apologies.
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  5. #5
    Senior Member Twig's Avatar
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    Quote Originally Posted by Plato View Post
    Label the triangle \Delta ABC. Any bit-triple will denote the direction the ant at the vertices goes. Example: \left( {0,1,1} \right) means that ant at A goes counter-clockwise while ants at B & C go clockwise. In that case ant A will collides with the ant at B. There are eight such triples. In how many will there be no collisions?
    I really liked this explanation, thanks!
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by Nupur View Post
    There are 3 ants at 3 corners of a triangle, they randomly start moving towards another corner.. what is the probability that they don't collide.

    All three should move in the same direction - clockwise or anticlockwise. Probability is 1/4.

    Can someone please explain the answer?
    If they are not all going in the same direction then a pair must be walking towards one another and so will collide, so to avoid collision they must all go in the same direction.

    Each ant has two choices of direction so the probability that they all go clockwise is (1/2)(1/2)(1/2)=1/8.

    Similarly the probability that they all go anti-clockwise is 1/8

    Hence the probability that they all go in the same direction is the probability that they all go clockwise plus the probability that they all go anti-clockwise= ..

    RonL
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