It has been shown that the classical binomial form of ascertainment, assuming a constant probability pi that any affected individual may become a proband for his pedigree, cannot describe a rather wide range of ascertainment procedures that might arise in practice. Some more general heuristic ascertainment formulas might then be preferred, and in this paper we consider the probabilistic basis for these formulas. We retain the binomial assumption of the classical scheme but allow the ascertainment probability to depend on the number of potential probands per pedigree. This probability can be expressed by an increasing or a decreasing function of that number. Various illustrations are given and situations where the "cooperative" binomial scheme should be valuable are discussed.