Unit Sphere Average Distance

Splitting up the spherical shell as infinitesimal circular discs that subtend an angle $\theta$ at the origin, the expected distance is $\begin{aligned} \tfrac{1}{4\pi}\int_0^\pi 2\pi \sin\theta \,d\theta \times 2\sin\tfrac12\theta & = \; \int_0^\pi 2\sin^2\tfrac12\theta \cos\tfrac12\theta\,d\theta \; = \; \Big[\tfrac43\sin^3\tfrac12\theta\Big]_0^\pi \; = \; \boxed{\tfrac43} \end{aligned}$