I randomly scatter $N$ points on a bounded rectangular plane $P$ with dimensions $A \times B$. To be more specific, for $N$ iterations, I choose a real number $x \in [0, A]$ and a real number $y \in [0, B]$, and place my point at the coordinates $(x, y)$.
Let $R$ be the radius of some circle I place somewhere on the plane $P$ that encloses no points. What is the probability distribution for the maximum permissible size of $R$?
Note - I specified $P$ to be a rectangular plane but, please, let it be any shape you wish if you can answer my question.