Two random arcs on a circle not overlapping

On the unit circle, colour red the counterclockwise arc from $(1,0)$ of angle $\theta_1$, where $\theta_1\sim\text{Unif}(0,2\pi)$. Independently draw $\alpha,\theta_2\sim\text{Unif}(0,2\pi)$, and colour blue the counterclockwise arc of angle $\theta_2$ that begins at the point at angle $\alpha$. What is the probability that the red and blue arcs do not intersect?

$1/6$. Measure angles as fractions of a full turn so all three variables are $\text{Unif}(0,1)$. The arcs are disjoint iff blue starts after red ends ($\theta_1<\alpha$) and does not wrap back to $0$ ($\theta_2<1-\alpha$). Conditioning on $\alpha$, the probability is $\int_0^1\alpha(1-\alpha)\,d\alpha=\tfrac16$.