Knowing that each point process (PP) is characterized by its void probabilities: $f(A)=P(N_A=0)$ as definition: the probability to have 0 points of the PP in the area $A$. And also, knowing that for a homogenous Poisson PP the void probability is $\exp(-\lambda A)$.
I'd like to know if there is an expression for the probability to have only 1 point in the area $A$ and/or a generic number $n$ of points: $P(N_A = 1), P(N_A = n)$ for the case of a Poisson PP?
I'm not looking for the immediate solution, I accept also preliminary inputs of the computation.
If $\Lambda$ is the intensity measure of the Poisson point process $P$, then the number of elements in a Borel set $B$ is Poisson distributed with parameter $\Lambda(B)$ (if $\Lambda(B) = \infty$, then there are almost surely infinitely many elements in $B$).
This is basically the definition of the Poisson point process.