The paper considers the best-choice secretary problem, with a known number, n, of applications, and a random, independent ‘freeze’ variable M, with known distribution. No hiring is possible after time M. The goal is to choose the best among the n applicants, where the decisions must be made depending only on the relative ranks of the applicants observed so far. A necessary and sufficient condition is given for the optimal rule to have the ‘simple’ structure: let k*-1 applicants pass, and stop with the first applicant (if any) from the k*th onward, who is better than all previous observed candidates. For uniform, geometric and Poisson freeze variables the optimal rules are simple. Some asymptotic results (as n⇒•), and minimax results are also discussed.
