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 math got too difficult for me, so I (shudder) looked it up.
This is turns out to be a special case of Sylvester's Four Point Theorem, and a complete answer can be found at:

http://mathworld.wolfram.com/SylvestersFour-PointProblem.html

It completely supports the assertions made by Charlie and ronen.

The exact answer is 11/36 (i.e. .30556), which is consistent with Charlie's simulation result of .305 and .306.

The site also gives values for other numbers of points, and for board shapes other than square.  If, for instance, the board is triangular, then the probability that the four points are convex is 1/3.

If the board is a random concave shape, then the probablity for four points is between  35/(12*(pi)^2)   (i.e., .29552)  and 1/3 (i.e., .33333)

Posted by Steve Herman on 2005-02-12 14:00:03