The probability of losing an allele when diploid genotypes are sampled

Abstract

Given the genotypic frequencies for a single gene locus with an arbitrary number of alleles in a hypothetically infinite population, the probability of losing at least one of the alleles in a sample of genotypes from this population is derived. The sensitivity of this probability to changes in the population frequencies of heterozygotes and of alleles has been investigated and has proved to be in accord with intuitive expectations. Least upper bounds for this probability, in cases where only the allelic frequencies or the minimum allelic frequency of alpha is known, have been found and the correspondence of these bounds to the situation of sampling of gametes rather than genotypes has been pointed out. It has been proved that the latter bound is also the least upper bound of the probability of losing at least one of the alleles with frequencies greater than or equal to alpha in a sample of genotypes or gametes if neither the number of alleles in the population nor the genotypic frequencies are known. This result has been used for the compilation of a table of the minimum sample sizes required to assure that all alleles with frequencies greater than or equal to alpha are detected with probability 1-sigma for the different values of alpha or sigma.