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).



The distance between the centroid of $P$ points and the centroid of a subset of the points
I might not be understanding the question, but the centroid is just the mean of the sample, so for mildly large samples from a mildly large set it will be (bivariate) normally distributed, and all the statistics are easy to compute.


