Chapter 4.4, Problem 109E

Buffon's Needle Experiment A horizontal plane is ruled with parallel lines 2 inches apart. A two-inch needle is tossed randomly onto the plane. The probability that the needle will touch a line is

$P=\frac{2}{\pi }{\displaystyle {\int }_0^{\pi /2}\sin }\theta d\theta$

where $\theta$ is the acute angle between the needle and any one of the parallel lines. Find this probability.