Intuitively, the Galois group (of a splitting field over $\mathbb Q$ of) a polynomial $f\in\mathbb Q[X]$ taken at random is most probably the full permutation group on the roots of $f$. This intuition can be made precise as follows.


*Elementary*: For inegers $d \geq 1$ and $N \geq 1$, consider the set $P\_{d,N}$ of polynomials in one variable of degree $\leq d$ whose coefficients have absolute value $\leq N$. The set $P\_{d,N}$ is finite, and we can consider the subset $Q\_{d,N}$ of those $f\in P\_{d,N}$ whose Galois group is the full symmetric group $S\_d$. Then, is it true that the ratio $|Q\_{d,N}|/|P\_{d,N}|$ goes to $1$ as $N$ goes to infinity? If so, are there good estimates for the speed of convergence?


*Less elementary, but still the same*: Let $k$ be a number field. For an integer $d\geq 1$, indentify $k$--points of the $d$--dimensional projective space $\mathbb P^d$ with nonzero polynomials of degree $\leq d$ over $k$ up to scalars. This way, we can speak about the Galois group of a point $f\in \mathbb P^d(k)$. For an integer $N$, let $p\_{d,N}$ be the number of points of $P^d(k)$ of height $\leq N$, and let $q\_{d,N}$ be the number of points of height $\leq N$ and with Galois group $S\_d$. Then, is it true that
$$\frac{q\_{d,N}}{p\_{d,N}}\to 1$$
as $N$ goes to infinity? And again, are there good estimates for the speed of convergence?


*Remark*: The subset of $\mathbb Q^{d+1} = \mathbb A^{d+1}(\mathbb Q)$ of those elements $(a\_0,\ldots, a\_d)$ such that the Galois group of the polynomial $f = a\_0+a\_1X+\cdots+a\_dX^d$ is $S\_d$ is Zariski dense in $\mathbb A^{d+1}$. This is so because the Galois group of the generic polynomial $f = t\_0+t\_1X+\cdots+t\_dX^d$ with indeterminates $t\_0,\ldots, t\_d$ over $\mathbb Q(t\_0,\ldots, t\_d)$ is $S\_d$ and the fact that $\mathbb Q$ (or any number field) is a Hilbertian field. 



The Galois group of a random polynomial
As Torsten remarks, the answer is yes and the result is well-known. P. X. Gallagher proved the following: If $E\_d(N)$ denotes the set of degree $d$ polynomials with coefficients $|a\_i|\le N$ with Galois group not equal to $S\_d$, then
$$
|E\_d(N)|\ll N^{d-1/2}\log N
$$This bound gives you what you want. There is a discussion and a list of references in this [sci.math post](http://www.math.niu.edu/~rusin/known-math/99/galois_rnd).


