The analysis of large-scale system behavior remains a challenge; fortunately, combinatorial testing techniques exist for assisting system testers. Covering arrays provide fault coverage, but do not have an efficient algorithm to determine location of faults. Detecting arrays provide both coverage and such an algorithm; however, there exist far fewer construction techniques for them than for covering arrays. In general, more tests are required for detecting arrays than for covering arrays. In this paper we concern ourselves with determining, asymptotically, how many such tests are required. We use the probabilistic method and the Lovász local lemma in conjunction with the Lambert W function to determine upper bounds on the sizes of detecting arrays.