If I have a disk domain $\mathcal{D} \subset \mathbb{R}^{2}$ of radius $r>1$, and two sets of points $a_1,\dots,a_n \in \mathcal{D}$ and $b_1,\dots,b_m \in \mathcal{D}$ distributed uniformly at random within the disk, what is the probability no pair of points $(a_i,b_j)$ are within a unit distance of each other?
I can put bounds on the probability by, for example: distributing $a_1,\dots,a_n$ randomly inside the disk, then drawing unit disks around each point, and asking what e.g. the area of the convex hull of the union of disks is, and then using the void probability i.e. the proportion of the time the $b$ points fall outside that area, since any point within is within a unit distance of an $a$ point. But can this be done more elegantly? Is an exact solution intractable, for example?