Probability Problem

Discussion in 'Probability' started by Rueful Rabbit, Oct 7, 2011.

  1. Hi All,

    A line one unit long is divided by 2 random points say A and B.
    What is the probability the longest of the 3 line segments is greater
    than twice the shortest?

    I've run a simulation on this and the answer in the ball park of 0.899
    assuming I've coded it correctly.
    However I've no joy on producing an analytical answer.

    Help appreciated.
     
    Rueful Rabbit, Oct 7, 2011
    #1
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  2. Rueful Rabbit

    Ray Koopman Guest

    Let {A,B} be iid Uniform[0,1], and let {U,V} = {min[A,B],max[A,B]}.
    Then the lengths of the line segments are {P,Q,R} = {U,V-U,1-V},
    and their joint distribution is the same as the joint distribution
    of {X,Y,Z}/(X+Y+Z), where X,Y,Z are iid Exponential variables.
    This is a (diffuse) Dirichlet distribution with parameters {1,1,1}:
    f[p,q,r] = 2; p,q,r > 0, p+q+r = 1.

    The most direct way to get the probability is by integrating.
    Mathematica gives

    In[1]:= 2 Integrate[Boole[ Max[p,q,1-p-q] > 2 Min[p,q,1-p-q] ],
    {p,0,1},{q,0,1-p}]
    Out[1]= 9/10
     
    Ray Koopman, Oct 8, 2011
    #2
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  3. Thanks Ray!
     
    rabbit.rueful, Oct 8, 2011
    #3
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