Put spheres uniformly at random all over $\mathbb{R}^3$, with density 1 sphere / unit cube. All spheres have the same radius $r$. What is the probability function $p(r)$, that that there is an infinite component, that is an infinite sequence of spheres such that the intersection of $s_n$ with $s_{n+1}$ is nonzero?
The distribution can be constructed like this: Partition $\mathbb{R}^3$ into cubes of sidelength $L$ and put $L^3$ spheres uniformly at random inside each. Then we take the distribution that arises as $L$ goes to infinity.
There exists $r_c$ such that $p(r)=0$ when $r<r_c$ and $p(r)=1$ for $r>r_c$. The value $p(r_c) \in \{0,1\}$ is not known. Neither is $r_c$.