A series of two-urn biased sampling problems

Much of the literature on matching problems in statistics has focused on single items chosen from independent, but fully overlapping sets. This paper considers a more general problem of sampling without replacement from partially overlapping sets and presents new theory on probability problems involving two urns and the selection of balls from these urns according to a biased without-replacement sampling scheme. The strength of the sampling bias is first considered as known, and then as unknown, with a discussion of how that strength may be assessed using observed data. Each scenario is illustrated with a numerical example, and the theory is applied to a gender bias problem in committee selection, and to a problem where competing retailers select products to place on promotion.