Probability Problem

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.

 

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

 

Thanks Ray!