Four pins are randomly nailed to a square board (according to a uniform probability distribution), and a rubber band is stretched completely around them to form a convex shape. What is the probability that the rubber band is in the shape of a triangle?
(In reply to
thoughts--and a simulation by Charlie)
This doesn't seem right to me, Charlie. Specifically, I doubt
that the cases of the first point placed, second point placed, etc. are
mutually exclusive. My concern is that if one of the points is
interior, then none of the others can be interior. So I don't
think that you can just take the expected area of a triangle and
multiply by 4.
I'm going to test this by taking a simpler case, where two of the
points are on the corners of a unit square, and the other two are
randomly placed inside. The area of the expected triangle is 1/4,
and twice this is 1/2. I think I can calculate the probability of
an interior point, and I suspect it won't be 1/2.
Or maybe my intuition is failing me. Stay tuned for results ...
re: thoughts--and a simulation
