Previous approaches to the ascertainment problem have been in terms of a registry-type situation in which the sampled structure is a random variable. However, with the exception of Bailey (1951), previous authors have conditioned on this structure. In extending the theory to pedigrees, a registry situation may also be considered, providing neither the structure nor any part of it (e.g., ages of individuals) are conditioned upon, but it may often be neither practicable nor relevant to do so. In many studies the sampling scheme may fix the number of pedigrees, analyzed, and classical "ascertainment probabilities" are no longer applicable. In an infinite population, in which the ascertainment of and phenotypic observations on, affect neither the distribution of other ascertainments or phenotypic distributions on other pedigrees and chance multiple ascertainments cannot occur, ascertainment corrections are relatively straightforward. However, effective population sizes with regard to particular rare traits may often be small, and the problem of ascertainment corrections in this case has not been fully analyzed, although some preliminary results are presented.