  We investigate the joint distribution of the vertex degrees in three models
of random bipartite graphs. Namely, we can choose each edge with a specified
probability, choose a specified number of edges, or specify the vertex degrees
in one of the two colour classes. This problem can alternatively be described
in terms of the row and sum columns of random binary matrix or the in-degrees
and out-degrees of a random digraph, in which case we can optionally forbid
loops. It can also be cast as a problem in random hypergraphs, or as a
classical occupancy, allocation, or coupon collection problem. In each case,
provided the two colour classes are not too different in size nor the number of
edges too low, we define a probability space based on independent binomial
variables and show that its probability masses asymptotically equal those of
the degrees in the graph model almost everywhere. The accuracy is sufficient to
asymptotically determine the expectation of any joint function of the degrees
whose maximum is at most polynomially greater than its expectation.
