  We analyze the probability that, for a fixed finite set of primes S, a
random, monic, degree n polynomial f(x) with integer coefficients in a box of
side B around 0 satisfies: (i) f(x) is irreducible over the rationals, with
splitting field over the rationals having Galois group $S_n$; (ii) the
polynomial discriminant Disc(f) is relatively prime to all primes in S; (iii)
f(x) has a prescribed splitting type at each prime p in S.
  The limit probabilities as $B \to \infty$ are described in terms of values of
a one-parameter family of measures on $S_n$, called splitting measures, with
parameter $z$ evaluated at the primes p in S. We study properties of these
measures. We deduce that there exist degree n extensions of the rationals with
Galois closure having Galois group $S_n$ with a given finite set of primes S
having given Artin symbols, with some restrictions on allowed Artin symbols for
p<n. We compare the distributions of these measures with distributions
formulated by Bhargava for splitting probabilities for a fixed prime $p$ in
such degree $n$ extensions ordered by size of discriminant, conditioned to be
relatively prime to $p$.
