If we choose randomly an infinite, countable set of disks in $\mathbb R^2$, what is the probability that intersection of every pair of disks from the set is an empty set?
EDIT: Because the problem in the above stated form is too hard to handle, or undecidable, or it may have many different solutions depending on the way we choose the centers and the radii I will state it with extra conditions.
First variant of the problem: Add to the original problem: "all disks have the same radius $R$".
Second variant of the problem: Add to the original problem "all disks have the same radius $R>1$ and every center has the coordinates $(x,y)$, where $x,y \in\mathbb Z$"
Question for the first and second variant of the problem: Is the problem solvable in at least one of these two variants or we need more extra conditions?
Sadly, some natural formulations of this problem are extremely difficult. Take a finite square with side-lengths $n$ and then make it periodic by identifying opposite sides, i.e. making it a torus. Then you can uniformly pick the centers of each circle within the rectangle. This is called the "Pennies on a carpet problem" or "2-D hard-spheres model." The problem is extremely difficult and not too much is known about it. You can find a bit more info here