Buffon's Needle

Theorem
Let a horizontal plane be divided into strips by a series of parallel lines a fixed distance apart, like floorboards.

Let a needle whose length equals the distance between the parallel lines be dropped onto the plane randomly from a random height.

Then the probability that the needle falls across one of the parallel lines is $\dfrac 2 \pi$.

Proof
For simplicity, consider $\mathbb R^2$ divided into strips by the lines $x=k$ for each integer $k$.

Then the needle would have length 1, the distance between the lines.

Define $\theta\in\left[-\dfrac\pi2,\dfrac\pi2\right)$ as the angle between the needle and the $x$-axis.

Then the horizontal component of length of the needle is $\cos \theta$ for each $\theta$.

Let $E$ be the event where the needle falls across the vertical lines,

and let $\Theta_\theta$ be the event where the angle between the needle and the $x$-axis is $\theta$.

By symmetry we may assume when the needle is dropped,

the end with the larger $x$-coordinate has $x$-coordinate $0\le x_n< 1$.

Then for each $\theta$, the needle falls across the line $x=0$ exactly when $0 \le x_n \le \cos \theta$.

Therefore the probability that this happens is $P(E\vert\Theta_\theta)=\dfrac{\cos \theta}{1}=\cos \theta$.

By considering $\theta$ as a continuous random variable,

Also known as
This problem is also known just as the needle problem.