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Math Help - Chords

  1. #1
    Member Veronica1999's Avatar
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    Chords

    Four distinct points A,B,C, and D, are to be selected from 1996 points evenly spaced around a circle. All quadruples are equally likely to be chosen. What is the probability that the chord AB intersects the chord CD?

    My answer 1/5 but it's wrong.
    The answer is 1/3. How could it be 1/3?

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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by Veronica1999 View Post
    Four distinct points A,B,C, and D, are to be selected from 1996 points evenly spaced around a circle. All quadruples are equally likely to be chosen. What is the probability that the chord AB intersects the chord CD?

    My answer 1/5 but it's wrong.
    The answer is 1/3. How could it be 1/3?
    Condiser the four points
    A,B,C,D The chord AB intersects CD if and only if they are arrainged around the circle as follow

    ACBD or ADBC

    These simplest way way to get the probability is There are
    4!=24 ways to arrange 4 letters
    As show above there are two ways to have the chords intersect, but there are 4 different arrangements eg ACBD \mapsto DACB \mapsto BDAC \mapsto CBDA

    You can do the same thing with the other to give 8. So the probability is

    \frac{8}{24}=\frac{1}{3}
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    It is 1/3.

    Hint:
    Compute:

    (The total number of quads)/(The total number of intersecting quads) = ...
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