Given n points drawn randomly on the circumference of a circle, what is the probability they will all be within any common semicircle?

Level: Senior

Let a_1, a_2, ..., a_n be n points randomly drawn. Let A_k, where k = 1, 2, ..., n, be the event that all the other n-1 points are counterclockwise from a_k and within an arclength of half a circle from a_k. Then p = P(A_k) = (1/2)^(n-1) = 1/2^(n-1) for k = 1, 2, ..., n. Since A_i and A_j, where 1 <= i < j <= n, are mutually exclusive, so the desired probability is P(A_1)+P(A_2)+...+P(A_n) = np = n/2^(n-1).