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 the real number plane $\R^2$ divided into strips by the lines $x = k$ for each integer $k$.

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

Define $\theta \in \hointr {-\dfrac \pi 2} {\dfrac \pi 2}$ 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,
 * $\Theta_\theta$ be the event where the angle between the needle and the $x$-axis is $\theta$.

Let the needle drop.

, let the end with the larger $x$-coordinate have $x$-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:
 * $\map \Pr {E \mid \Theta_\theta} = \dfrac {\cos \theta} 1 = \cos \theta$

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

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