  Fine's theorem concerns the question of determining the conditions under
which a certain set of probabilities for pairs of four bivalent quantities may
be taken to be the marginals of an underlying probability distribution. The
eight CHSH inequalities are well-known to be necessary conditions, but Fine's
theorem is the striking result that they are also a sufficient condition. It
has application to the question of finding a local hidden variables theory for
measurements of pairs of spins for a system in an EPRB state. Here we present
two simple and self-contained proofs of Fine's theorem in which the origins of
this non-obvious result can be easily seen. The first is a physically motivated
proof which simply notes that this matching problem is solved using a local
hidden variables model given by Peres. The second is a straightforward
algebraic proof which uses a representation of the probabilities in terms of
correlation functions and takes advantage of certain simplifications naturally
arising in that representation. A third, unsuccessful attempt at a proof,
involving the maximum entropy technique is also briefly described
