Imagine I have a set of $(s_1,...,s_N) \in S$ Brownian particles in a box of sidelength $L$, each with the same coefficient of diffusion $D$. We fix one particle at the center of the box, and draw a sphere of radius $r$ about it. Equivalently, we draw a sphere of the same radius around one of the diffusing particles without bothering to fix the particle in place. Fick's law can be used to tell us the diffusive flux through this sphere as a function of time.

Let $V$ be some duration of time in seconds. What is the probability that the sphere only contains the one Brownian particle at its center for a continuous stretch of $q \leq R$ seconds over this time interval?