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It is well-known that for any finite group G, there exists a closed 3-manifold M with G as a quotient of pi_1(M). However, we can ask more detailed questions about the possible finite quotients of 3-manifold groups, e.g. if G and H are finite groups, does there exist a 3-manifold group with G as a quotient but not H as a quotient?  We give an answer to all such questions.  To prove existence of 3-manifolds with certain finite quotients but not others, we use a probabilistic method, by first proving a formula for the distribution of the (profinite completion of) the fundamental group of a random 3-manifold in the Dunfield-Thurston model of random Heegard splittings as the genus goes to infinity.  This is joint work with Will Sawin.