1. Consider a population of size N, of which piN items have the characteristic A, of which, in turn, p2N also have the characteristic B; i.e., the proportion of A-items which are also B-items is P2/Pl. For example, pi may be the proportion of individuals eligible to retire among a group of individuals covered by a retirement program at the beginning of the year and p2/pl the proportion of eligibles who retire during the year, or pi may be the proportion of families in a given income class among a group of families in all income classes and P2/Pl the proportion of families in the specified income class who own their own homes. A straightforward problem is one of estimating P2/Pl from a simple random sample of size n drawn from N. The problem as stated in the title is, of course, equivalent to one of estimating the ratio of the overlap between two overlapping classes to one of the classes. Sometimes the actual value of pi or P2 is known beforehand. On the theory that it is always to use a known than an estimated datum, one might be tempted to estimate p2/pl by estimating the unknown proportion from the sample and combining that estimate with the known proportion. This, however, is not necessarily the best course to follow. If we consider the better of two estimates to be the one with the smaller coefficient of variation, it can be shown, for a sufficiently large sample, that (a) even if P2 is known, a estimate of P2/Pl (or of P1/P2) can be obtained by estimating both proportions from the sample if p2>pj/(2-pl), or, alternatively, if pl<2p2/(l+p2), and (b) even if pi is known, a estimate of P2/p1 (or of P1/P2) can be obtained by estimating both proportions from the sample, regardless of the values of pi and P2. The criterion in (a) cannot be applied exactly in practice, since pi is not known, but its sample value can almost always be satisfactorily substituted for the universe value in the inequalities. By way of illustration, suppose that, on the basis of a simple random sample taken of 100,000 employees covered by a pension program at the beginning of a year, it is estimated that 5,000 were eligible to retire during the year and that 3,000 of these did retire. Suppose,further, that the exact number of those who retired is known, from another