Striking applications of linearity of expectation

Buffon’s needle: rule a surface with parallel lines a distance $d$ apart. What is the probability that a randomly dropped needle of length $\ell\leq d$ crosses a line?

Consider dropping any (continuous) curve of length $\ell$ onto the surface. Imagine dividing up the curve into $N$ straight line segments, each of length $\ell/N$. Let $X_i$ be the indicator for the $i$-th segment crossing a line. Then if $X$ is the total number of times the curve crosses a line,
$$\mathbb E[X]=\mathbb E\left[\sum X_i\right]=\sum\mathbb E[X_i]=N\cdot\mathbb E[X_1].$$
That is to say, the expected number of crossings is proportional to the length of the curve (and independent of the shape).

Now we need to fix the constant of proportionality. Take the curve to be a circle of diameter $d$. Almost surely, this curve will cross a line twice. The length of the circle is $\pi d$, so a curve of length $\ell$ crosses a line $\frac{2\ell}{\pi d}$ times.

Now observe that a straight needle of length $\ell\leq d$ can cross a line either $0$ or $1$ times. So the probability it crosses a line is precisely this expectation value $\frac{2\ell}{\pi d}$.