The Probability of Pairwise Shared Ancestry and the Expected Number of Pairs of k-th Cousins in a Population Sample

An analytical solution is derived for the probability that a random pair of individuals from a panmictic population of size N will share ancestors who lived G generations previously. The analysis is extended to obtain 1) the probability that a sample of size s will contain at least one pair of (G - 1)th cousins; and 2) the expected number of pairs of (G - 1)th cousins in that sample. Solutions are given for both monogamous and promiscuous (non-monogamous) cases. Simulation results for a population size of N = 20,000 closely approximate the analytical expectations. Simulation results also agree very well with previously derived expectations for the proportion of unrelated individuals in a sample. The analysis is broadly consistent with genetic estimates of relatedness among a sample of 406 Danish school children, but suggests that a different genetic study of a heterogenous sample of Europeans overestimates the frequency of cousin pairs by as much as one order of magnitude.