Topics include: ways modern statistical procedures can yield estimates of pi more precisely than the original Buffon procedure traditionally used; the question of density and measure for random geometric elements that leave probability and expectation statements invariant under translation and rotation; the number of random line intersections in a plane and their angles of intersection; developments due to W. L. Stevens's ingenious solution for evaluating the probability that n random arcs of size a cover a unit circumference completely; the development of M. W. Crofton's mean value theorem and its applications in classical problems; and an interesting problem in geometrical probability presented by a karyograph.