  Let S and T be vectors of positive integers with the same sum. We study the
uniform distribution on the space of simple bipartite graphs with degree
sequence S in one part and T in the other; equivalently, binary matrices with
row sums S and column sums T. In particular, we find precise formulae for the
probabilities that a given bipartite graph is edge-disjoint from, a subgraph
of, or an induced subgraph of a random graph in the class. We also give similar
formulae for the uniform distribution on the set of simple directed graphs with
out-degrees S and in-degrees T. In each case, the graphs or digraphs are
required to be sufficiently dense, with the degrees varying within certain
limits, and the subgraphs are required to be sufficiently sparse. Previous
results were restricted to spaces of sparse graphs. Our theorems are based on
an enumeration of bipartite graphs avoiding a given set of edges, proved by
multidimensional complex integration. As a sample application, we determine the
expected permanent of a random binary matrix with row sums S and column sums T.
