[ExI] isaac's number

[spike]

Cool!  My son posed a mathematical puzzle, the answer to which we are
calling Isaac's number until we find someone else who has posed the problem
and answered it:

 

Generate three random points on a square plane to create a triangle.  What
is the probability that a fourth randomly-generated point will be inside the
triangle?

 

He and I wrote sims and both get the same answer to four significant digits:
0.0764 but neither of us know how to find it in closed form.  My intuition
suggested that number should have been 0.125 but it differs from our
sim-generated Isaac's number by about phi.  Why phi?  That hasta be a
coincidence, ja?  Come on math geeks, let us reason together, shall we?

 

spike

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