Suppose I have a set $A \subset \mathbb{N}$ and I want to pick elements at random and check whether they belong to $A$ or not.
I know that the probability for a given $k$-digit number $n \in \mathbb{N}$ to belong to $A$ is given by some $p_k$ that depends on the number of digits $k$ of $n$, for example we could have $p_k=\frac{1}{k}$ or $p_k = 10^{-k}$.
If I pick at random a $1$-digit integer, then a $2$-digits integer, and so on, how can I compute the probability, that (in)finitely many of the picked numbers belong to $A$?
The details about picking numbers with certain numbers of digits do not really affect this answer. All we need for a complete answer are infinitely many events that are at least pairwise independent (I imagine your events for choosing the numbers were intended to be mutually independent.).
By the Borel-Cantelli Lemma, if the sum of the $p_k$ is finite (as with $p_k=\dfrac{1}{9*10^{k-1}}\approx10^{-k}$), then the probability of infinitely many events occurring (numbers belonging to $A$) is $0$. And (assuming pairwise independence), if the sum of the $p_k$ is infinite (as with $p_k\approx\frac1k$), then the probability is $1$ by a sort of converse result.