Threshold sampling

We consider the problem of sampling elements with some desired property from a large set, without testing the property of interest, but with the (probabilistic) assurance to have at least one match among the random sample. Like in ranked set sampling (RSS), we consider an infinite population under study, whose properties of interest are too expensive and/or time-consuming to measure. Unlike RSS, we are void of a ranking mechanism, so our sampling is done entirely blind. We show how it is nonetheless doable to assure, with controllably large likelihood, to either have at least one of the interesting elements in a random sample, or, contrarily, sample with the likewise assurance of not having one of the interesting elements in the sample. Our technique utilizes density bounds for distributions and threshold functions from random graph theory.