If Buffon's needle has length … instead of 2, find the probability…

(A classic way to compute $\pi$) A $2^{\prime \prime }$ needle is tossed onto a floor with boards $2^{\prime \prime }$ wide. Find the probability of falling across a crack. (This happens when $\cos \theta >y$ = distance from midpoint of needle to nearest crack. In the rectangle $0\le \theta \le \pi /2$, $0\le y\le 1$, shade the part where $\cos \theta >y$ and find the fraction of area that is shaded.)
30 If Buffon's needle has length $2x$ instead of 2, find the probability $P(x)$ of falling across the same cracks.

If Buffon's needle has length $2x$ instead of 2, find the probability $P(x)$ of falling across the same cracks.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §5.6 Properties of the Integral and Average Value, #30