Chapter 6, Problem 6.16P

Suppose that n points are independently chosen at random on the circumference of a circle, and we want the probability that they all lie in some semicircle. That is, we want the probability that there is a line passing through the center of the circle such that all the points are on one side of that line, as shown in the following diagram:

Let $P_1,\mathrm{..}\mathrm{..}{P}_n$ denote the n points. Let A denote the event that all the points are contained in some semicircle, and let $A_i$ be the event that all the points lie in the semicircle beginning at the point $P_i$ and going clockwise for $180°,i=1,\mathrm{...},n$.

a. Express A in terms of the $A_i$.

b. Are the $A_i$ mutually exclusive?

c. Find P(A).