If we suppose that we have a polynomial $q(x)$ of the following form:
$q(x) = \sum_{i=0}^N{c_i x^i} \text{ where } c_i=0 \text{ or } c_i=1$
In other words, if we are given a polynomial with binary coefficients (either 0 or 1), what is the probability that $q(x) \bmod p \equiv 0$ where $p$ is prime? There is a twist. We set $x$ equal to 1. That's the probability I'm after. It should be pretty simple.
Now, what I'm really after is a proof that as $N$ tends to infinity, this probability tends to $1/p$ if we set $x=1$.. I'd be delighted if someone could prove this. However, I'd still like to know the probability for any $N$. Additionally, I'm also interested in the cases where we set $x$ equal to a natural less than $p$.