  We consider random lines in $\mathbb{R}^3$ (random with respect to the
kinematic measure) and how they intersect $\mathbb{S}^2$. It is known that the
entry point and the exit point behave like \textit{independent} uniformly
distributed random variables. We give a new proof using bilinear integral
geometry and use this approach to show that this property is extremely rare: if
$K \subset \mathbb{R}^n$ is a bounded, convex domain with smooth boundary with
this property (i.e., the intersection points with a random line are
independent), then $n=3$ and $K$ is a ball.
