Imagine I place discs of radius $r$ on a two-dimensional plane, selecting their positions with uniform probability across the surface of the plane, and stop when I reach a disc density $\rho$. As a function of $r$, what is the probability that a disc chosen with uniform random probability overlaps another as a function of $\rho$, and is there a closed-form expression for this probability? Is there a simple generalization for this problem to higher dimensions, say for three-dimensional spheres placed with uniform probability in a volume $V$ until a certain density is achieved?



Probability of disc-disc overlap for discs placed with uniform probability on a surface until a density $\rho$ is achieved
There's no such thing as "uniform probability across the surface of the plane", only uniform probability on a set of finite measure. But perhaps what you're looking for is a spatial Poisson process (for the centres of the discs) of density $\rho$. Then if you pick a random point, the probability that a disc centred there will
overlap one or more of the discs of the process, i.e. that one or more 
points of the process is within distance $2 r$ of the new point, is 
$1 - e^{-4 \pi r^2 \rho}$. The generalization to $d$ dimensions is easy. 


