  In this paper, we study random instances of the classical marginal problem.
We encode the problem in a graph, where the vertices have assigned fixed binary
probability distributions, and edges have assigned random bivariate
distributions having the incident vertex distributions as marginals. We provide
estimates on the probability that a joint distribution on the graph exists,
having the bivariate edge distributions as marginals. Our study is motivated by
Fine's theorem in quantum mechanics. We study in great detail the graphs
corresponding to CHSH and Bell-Wigner scenarios providing rations of volumes
between the local and non-signaling polytopes.
