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A: Given two random points x and y on the interval 0..1, what is the average
size of the smallest of the three resulting intervals?


Q: You could make a graph of the size of the smallest interval versus x
and y.  We know that the height of the graph is 0 along all the edges
of the unit square where it is defined and on the line x=y.  It
achieves its maximum of 1/3 at (1/3,2/3) and (2/3,1/3).  In between
these positions the surface forms a series of planes.  Thus the volume
under it consists of 2 pyramids each with an altitude of 1/3 and an
(isosceles triangular) base of area 1/2, yielding a total volume of
1/9.
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