Imagine I have an $(p_1, ..., p_N) \in P$ points, on a two-dimensional plane, patterned in a rectangular or hexagonal lattice arrangement in a circle of radius $R_c$, with a spacing between the points of $r_s$.

Let $C_P$ be the centroid of the $P$ points. If I randomly select some subset of $k$ points from $P$, and I compute the centroid of these $k$ points, $C_k$, what is the probability that the distance between $C_p$ and $C_k$ is $\leq D$?

Update: I have specified that the $P$ points in the circle are in a rectangular or hexagonal lattice arrangement (whichever is easiest to analyze).