  For a number field K and a finite abelian group G, we determine the
probabilities of various local completions of a random G-extension of K when
extensions are ordered by conductor. In particular, for a fixed prime p of K,
we determine the probability that p splits into r primes in a random
G-extension of K that is unramified at p. We find that these probabilities are
nicely behaved and mostly independent. This is in analogy to Chebotarev's
density theorem, which gives the probability that in a fixed extension a random
prime of K splits into r primes in the extension. We also give the asymptotics
for the number of G-extensions with bounded conductor. In fact, we give a class
of extension invariants, including conductor, for which we obtain the same
counting and probabilistic results. In contrast, we prove that that neither the
analogy with the Chebotarev probabilities nor the independence of probabilities
holds when extensions are ordered by discriminant.
