Let Q denote the unit square, [0,1] X [0,1]. Problem 1: Given two random points in Q what is the probability that the line determined by these two points will intersect two adjacent sides of Q? Problem 2: Given four random points in Q what is the probability that the quadrilateral determined by these four points is convex? A theoretical answer to the first problem is obtained and confirmed by computer simulation. An answer to the second problem is obtained by computer simulation.