ABSTRACT
Pick three points on the perimeter of the unit circle around the origin O, independently with uniform distribution. What is the probability that their convex hull contains O? There is a short and sweet argument that goes back at least to the sixties (see Wendel [3]), which shows that the answer is 1/4. For any point x on the circle, let −x denote the point diametrically opposite to x. For any distinct points x 1, x 2, and x 3 on the circle, consider the unordered triples T={ε 1 x 1,ε 2 x 2,ε 3 x 3}, where each ε i =+1 or −1. Observe that out of these 8 triples precisely 2 induce a triangle which contains the origin in its interior, and the claim readily follows.
