Buffon's needle problem

Problem

A floor has parallel lines at distance $t > 0$ apart. A needle of length $l$ with $0 < l \le t$ falls at random on the floor. "At random" means this. The distance $x$ of the centre of the needle from the nearest line is uniform on $[0, t/2]$. The acute angle $\theta$ between the needle and the lines is uniform on $[0, \pi/2]$. The variables $x$ and $\theta$ are independent. Find the probability that the needle crosses a line.

Remarks

The answer is $\dfrac{2l}{\pi t}$. The needle crosses a line exactly when $x \le \frac{l}{2} \sin \theta$, and the integral of this region gives the answer. For a needle with $l = t$, the probability is $2/\pi \approx 0.637$. The answer contains $\pi$, so repeated throws give a random estimate of $\pi$. For a long needle with $l > t$, the answer is a different formula.

History

Georges-Louis Leclerc, Comte de Buffon, posed the problem in the 18th century. It was one of the first problems of geometric probability.

Explanations

L1 human

The needle crosses a line when it lands close to a line and at a steep angle. The chance is the area of the "good" pairs (distance, angle) divided by the area of all pairs. A sine function appears in this area. That is how $\pi$ comes into the answer.