Suppose we have a floor made of parallel strips of wood, each with width 3, and we drop a needle of length 1 onto the floor. The aim of this exercise is to find the probability that the needle will lie across a line between two strips.
(a) Assume that the strips of wood lie in the east-west direction. We call one end of the needle the head and the other end the tip. Let X denote the distance of head from the closest separating line to the south of the head. Let Y denote the angle of the needle measured in radians (let's say Y = 0 if the needle is parallel to the lines and the tip points to the east). What is the natural choice of the joint p.d.f. of (X, Y ) if we want to model a randomly dropped needle?
(b) The possible outcomes of (X,Y) form a rectangle. Describe (and draw) the subset A of this rectangle whose points correspond to a needle position that crosses a line!
(c) Calculate the probability that the needle will lie across a line between two strips.