Based on the graphic in the MathWorld article on the quintic equation it seems very likely that following statement is either true or trivially false in a way that can be easily remedied by adding an additional hypothesis:
Let $A_N$ be the set of degree $n$ polynomials $f(x)$ with integer coefficients $a_0, a_1, ... a_n$ such that $|a_i| < N$. Let $S_N$ be the fraction of these that have Galois group $S_n$. Then as $N \rightarrow \infty$ , $S_N \rightarrow 1$.
It also seems likely that this result follows from well known results: how does one prove it?