Consider a disk centered at O with radius R. Suppose that n ≥ 3 points P 1 , P 2 , ... , P n are independently placed at random inside the disk. Find the probability that all these points are...

Consider a disk centered at O with radius R. Suppose that n ≥ 3 points P₁, P₂, ... , P_n
are independently placed at random inside the disk. Find the probability that all these points are contained in a closed semicircular disk. Hint: For each 1 ≤ i ≤ n, let Ai be the endpoint of the radius through Pi and let Bi be the corresponding antipodal point. Let Di be the closed semicircular disk OA_iB_i
with negatively (clockwise) oriented boundary. Note that there is at most one D_i, 1 ≤ i ≤ n, that contains all the P_i’s.