  For n>=1 let X_n be a vector of n independent Bernoulli random variables. We
assume that X_n consists of M "blocks" such that the Bernoulli random variables
in block i have success probability p_i. Here M does not depend on n and the
size of each block is essentially linear in n. Let X'_n be a random vector
having the conditional distribution of X_n, conditioned on the total number of
successes being at least k_n, where k_n is also essentially linear in n. Define
Y'_n similarly, but with success probabilities q_i>=p_i. We prove that the law
of X'_n converges weakly to a distribution that we can describe precisely. We
then prove that sup Pr(X'_n <= Y'_n) converges to a constant, where the
supremum is taken over all possible couplings of X'_n and Y'_n. This constant
is expressed explicitly in terms of the parameters of the system.
