2.4 Buffon’s needle problem.

Suppose that a needle is tossed at random onto a plane ruled with parallel lines a distance L apart, where by a “needle” we mean a line segment of length l ≤ L.

What is the probability of the needle intersecting one of the parallel lines?

Hint: Consider the angle that is made by the needle with the parallel lines as a random variable α uniformly distributed in the interval [0, 2π] and the position of the midpoint of the needle as another random variable ξ also uniform on the interval [0, L]. Then express the condition “needle intersects the parallel lines” in terms of the position of the midpoint of the needle and the angle α. Do a calculation similar to Example 2.5.