4.3.11 problem 50

Let $C_1$ be a random chord that spans a minor arch of length $x$ on a circle of radius $r$. To generate a chord $C_2$, with endponts $A$ and $B$, such that $C_2$ intersects $C_1$, either $A$ is on the minor arch and $B$ is on the major arch, or $A$ is on the major arch and $B$ is on the minor arch.

Let $I_x$ be the indicator variable for $C_2$ intersecting $C_1$ when the minor arch generated by $C_1$ has length $x$. Then, $P(I_x=1)=2\frac{x}{2\pi \ast r}\ast \frac{2\pi \ast r-x}{2\pi \ast r}$.

Since the length of the minor arch $x$ generated by $C_1$ can span from $0$ to $2\pi \ast r$, we integrate $P(I_x=1)$.