I already posted this question at MSE here, but since it received no answer or comment so far I cross-post it here.

It is well-known that if one considers a “random” monic polynomial of fixed degree, say $X^n+\sum_{k=0}^{n-1}a_kX^k$ where $(a_0,a_1,\ldots, a_n)$ is drawn from the discrete uniform distribution on $[-N,N]^{n+1}$, then this polynomial will be irreducible and have Galois group $S_n$ “almost surely”, i.e. the probability of this event tends to $1$ when $N\to \infty$.

Now, suppose one considers two random monic polynomials $P=X^n+\sum_{k=0}^{n-1}a_kX^k$ and $Q=X^m+\sum_{k=0}^{n-1}b_kX^k$ where $(a_0,a_1,\ldots, a_n,b_0,\ldots,b_m)$ is drawn from the discrete uniform distribution on $[-N,N]^{n+m+2}$. Is it also true that for any root $\alpha$ of $P$ and any root $\beta$ of $Q$, the extensions ${\mathbb Q}(\alpha)$ and ${\mathbb Q}(\beta)$ will be linearly disjoint over $\mathbb Q$ almost surely?