I drop two arcs on the unit circle of length $l_1$ and length $l_2$. Their starting location is random. What is their expected overlap as a function of $l_1$ and $l_2$?
This is related to:
Probability of overlap of random intervals dropped on unit circle
But not the quite the same.
Let the arcs be $A_1$ and $A_2$, and let $p$ be any given point on the circle. What is the probability that $p$ is in the intersection? The events $p \in A_1$ and $p \in A_2$ are independent with probabilities $l_1/(2\pi)$ and $l_2/(2 \pi)$ respectively, and so the probability of their intersection is $l_1 l_2/(4 \pi^2)$. By Fubini's theorem, the expected measure of intersection is $$\mathbb E \left[ \int_0^{2\pi} dp \; 1_{p \in A_1} 1_{p \in A_2}\right] = \int_0^{2\pi} dp\; l_1 l_2/(4 \pi^2) = l_1 l_2/(2 \pi)$$
