A random line intersects $\mathbb{S}^2$ in two probabilistically independent locations

We consider random lines in $\mathbb{R}^3$ (random with respect to the kinematic measure) and how they intersect $\mathbb{S}^2$. We prove that the entry point and the exit point behave like independent uniformly distributed random variables. This property is very rare; we prove that if $K \subset \mathbb{R}^n$ is a bounded, convex domain with smooth boundary satisfying this property (i.e., the intersection points with a random line are independent), then $n=3$ and $K$ is a ball.