# Distance between poisson points in two disjoint unit discs

Two disjoint unit discs $D_1$ and $D_2$. Inside them there are random poisson points with intensity $\lambda$. For a given real $r>0$, what is the probability that there exist a poisson point $x_1\in D_1$ and a poisson point $x_2\in D_2$ such that $\|x_1-x_2\|_2 \le r$? Couldn't find any existing results on this.

-
If $M$ is the $4$-dimensional Lebesgue measure of $\{(x,y) \in D_1 \times D_2: \|x - y\|_2 \le r\}$, then the number of such pairs of points is a Poisson random variable with parameter $M \lambda^2$, so the probability that there is at least one is $1 - \exp(-M \lambda^2)$.
But how to compute $M$? –  user32828 Apr 7 at 11:58
Integrate over $x \in D_1$ the area of the intersection of $D_2$ with the disk of radius $r$ about $x$. I doubt that you'll end up with a closed-form expression. –  Robert Israel Apr 8 at 18:41