Chapter 9.2, Problem 6CP

One of the best-known Monte Carlo problems is the Buffon needle. If a needle is dropped on a floor painted with black and white stripes, each the same width as the length of the needle, then the probability is $2/\pi$ that the needle will straddle both colors. (a) Prove this result analytically. Consider the distance d of the needles midpoint to the nearest edge, and its angle $\theta$

with the stripes. Express the probability as a simple integral. (b) Design a Monte Carlo Type 2 simulation that approximates the probability, and carry it out with $n={10}^6$ pseudo-random pairs $(d,\theta )$.