Exactly one overlap between two random circle arcs

On the unit circle, draw a red arc that starts at the point $(1,0)$ and runs counterclockwise through an angle $\theta_1\sim\text{Unif}(0,2\pi)$. Independently pick $\alpha,\theta_2\sim\text{Unif}(0,2\pi)$ and draw a blue arc that starts at angle $\alpha$ (measured counterclockwise from $(1,0)$) and runs counterclockwise through an angle $\theta_2$, wrapping past $(1,0)$ if necessary. Wherever the arcs overlap the colour is purple. Assuming some purple appears, how likely is it that the purple part forms a single connected piece?

$4/5$. Rescale angles to $[0,1)$. The arcs miss each other with probability $\tfrac16$, and they overlap in two separate pieces exactly when the blue arc starts inside the red one and wraps past $(1,0)$, i.e. $\alpha<\theta_1$ and $\alpha+\theta_2>1$, which has probability $\int_0^1 x(1-x)\,dx=\tfrac16$. So $P(\text{one piece}\mid\text{overlap})=\tfrac{2/3}{5/6}=\tfrac45$.