# Probability Problem

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

1. ### Rueful RabbitGuest

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

2. ### Ray KoopmanGuest

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