I understand that a generic $G$-polynomial $f(t_1,...,t_n)[X]$ over field $k$ has Galois group $G$ over $k(t_1,...,t_n)$. And basically any $G$ extension of $k$ should be generated by a realization of $f$.(even a bit stronger but that is not the point here).

Now as much as I understand, our motivation for hunting these polynomials is that in real (constructive) life, we would like to plug random elements of $k$ into $t_1,...,t_n$ and get a $G$-extension. However, it's obvious that the definition doesn't guarantee it. For example as a trivial failure, we know that $X^n + t_1X^{n-1} + \cdots + t_n$ is generic for $S_n$, but not all values for $t_1, ..., t_n$ (basically all polynomials) lead to an $S_n$-extension.

So, basically, my question is this: what is the constructive value of the definition of generic polynomial. Is there any (although I know I'm saying nonsense) high probabilistic/statistic success rate in getting a $G$-extension when a random realization is chosen. Is there some kind of definition of "odd" that says those times that we don't get a $G$-extension are somehow odd and not normal?