To clarify the notation, for a finite set $K$, let $s(K)$ be the sum of the elements of $K$.
$\text{My question}$
Quite straight forward, I want to ask this:
Given $A\subset\mathbb{N}$ such that $A$ is finite, what the the probability that there exists $B\subset A$ such that $s(B)$ is a prime number.
$\text{Motivation}$
This is mainly inspired by the fact that the probability of a number, $N$, being a prime is $\sim\frac{1}{\log{N}}$ and the fact that in a recent post, I talked about the probability of the sum of $n$ randomly chosen integers being divisible by $p$, a prime (which was later on generalized).
To be honest, I have almost no knowledge in probability or advanced number theory whatsoever, so the inly background I can provide you with is this idea I got:
When I first thought of the probability of a number being a prime, I had this train of thought (more or less, this is a summary):
So i have $\pi(n)$ favorable cases and $n$ total cases, as $n$ goes to infinity, so the probability would be: $$\lim_{n\to\infty}\frac{\pi(n)}{n}=\lim_{n\to\infty}\frac{1}{\log{n}}$$ But that gives us $0$, which is a contradiction. So then I thought of something very similar but simpler: instead of having $\pi(m)$ favorable cases and $m$ total cases as $m$ gets arbitrarely big, we can just have $n$ favorable cases and $p_n$ total cases, because $n\leq p_n$. So the probability now is $$\sim\frac{n}{p_n}\sim\frac{n}{n\log{n}}=\frac{1}{\log{n}}$$
But after doing this, I thought I could use an almost identical approach to see what is the probability for a number $n$ such that $n+c$ is a prime (for a given constant $c$), but I am not sure this actually works.
However, I am not sure how to generalize this to sums or sets. To sum up, I would appreciate any help or ideas.
Thank you for reading.