paper came from data acquired during quarantine searches, where the construction of confidence intervals for the sum, rather than difference, of two binomial proportions was of interest. In detail, shipping containers arriving at a frontier contained a certain number, $N$ say, of consignments. Some of the consignments might be clean, but others could contain pests which needed to be detected and removed to prevent their introduction to the environment. To reduce the costs associated with inspection, quarantine services usually inspect only $n_1<N$ consignments. Consignments are assumed to be contaminated with probability $p_1$, and the number, $n_1 {\bar X}_1$ say, of contaminated consignments found after routine (but incomplete) inspection of the items in each of $n_1$ consignments is assumed to follow a binomial distribution. Contaminated consignments are then “cleaned,” and the members of a subsample of $n_2$ of them are reinspected. (The data gathered in this way comprise a “leakage survey.”) The number of items, $n_2 {\bar X}_2$, still found contaminated (for example, contaminated with a different kind of pest) are assumed to follow a binomial distribution with parameters $n_2$ and $p_2$, and typically it is argued that ${\bar X}_1$ and ${\bar X}_2$ are independent. An estimator of the proportion of 