Reliability predictions for systems exhibiting few, if any, failures require the use of all available information. The Bayes equation incorporates prior engineering information with test data to provide statistically improved posterior estimates. Classical results agree with those obtained from the Bayes equation by using no prior information. For the case of failure-on-demand, this is equivalent to assuming a 50% mean failure probability for the prior information—hardly an appropriate estimate for a reliable system such as a reactor scram system. The method of Bayes conjugates applied to the cases of aging failure and failure-on-demand yields formulas for calculating mean, standard deviation, and confidence values. Various methods for incorporating prior information are possible. For example, calculating scram failure probabilities by incorporating prior information obtained from fault tree analysis of a scram system with historical test data indicates a mean scram failure probability of ∼8 × 10−6 per demand.