The Optimal Choice of a Subset of a Population

We check a ranked population in random order. At each step we either accept or reject the observed sample. As a result we get a partition of the population into two parts: accepted and rejected. We are interested only in the case when all accepted samples are better than all rejected. Under this condition, we maximize the probability to obtain a fixed size k of the accepted part. An arbitrary gain function φ(k) is also considered.