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?
The insight is to realize that if you put 3 pins in the board, the probability is not just the probability that the fourth pin falls within the triangle formed by the first three. If you draw the three lines formed by the 3 pins (extend the sides of the triangles to the edge of the board), they form 3 additional regions in which you can put a pin that will create a larger triangle. This is much easier to show graphically than describe, but there it is.
If you sum the areas of these regions, the triangle and the three additional regions, that is the area to divide by the total board area.
I did a numerical solution of this and got an average percentage of almost exactly 35%. The density function is interesting, but makes some sense. Unfortunately, I don't really have an intuitive feel for this problem.
Numerical solution
