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?



Why do generic polynomials work in reality?
Adding unto Boyarsky's answer: Stephen Cohen has given [quantative bounds](http://www.ams.org/mathscinet-getitem?mr=628276) for how often generic polynomials work. If I've skimmed his paper correctly, when the coefficients are integers chosen from the interval $[-N, N]$, the probability that the Galois group comes out wrong is $O(N^{-1/2} \log N)$, with an explicitly computable constant which depends on the group and the precise parameterization being used.


