Security system designers favor worst-case security measures, such as those derived from differential privacy, due to the strong guarantees they provide. These guarantees, on the downside, result on high penalties on the system's performance. In this paper, we study the Bayes security measure. This measure quantifies the expected advantage over random guessing of an adversary that observes the output of a mechanism. We show that the minimizer of this measure, which indicates its security lower bound, i) is independent from the prior on the secrets, ii) can be estimated efficiently in black-box scenarios, and iii) it enables system designers to find low-risk security parameters without hurting utility. We provide a thorough comparison with respect to well-known measures, identifying the scenarios where our measure is advantageous for designers, which we illustrate empirically on relevant security and privacy problems.