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In this report an analagous Bayesian approach is given to the classical methodology for calculating lower bounds on system reliability as formulated in th Maximus Handbook of February 1980. For ease of comparison the Maximus format is adhered to as far as possible. All theoretical work is given in appendices, as is the description of the computer programs that were developed to facilitate calculations. In addition to the pass/fail test data considered in Maximus the case of exponential times to failure of components is treated. Keywords: Probability distributions; Likelihood functions; Random variables.