An engineer is studying the reliability of a product by performing a sequence of n trials. Reliability is defined as the probability of success. In each trial,the product succeeds with probability p and fails with probability 1 ? p. The trials are conditionally independent given p. Here p is unknown (else the study would be unnecessary!). The engineer takes a Bayesian approach, with p ? Unif(0, 1) as prior. Let r be a desired reliability level and c be the corresponding confidence level, in the sense that, given the data, the probability is c that the true reliability p is at least r. For example, if r = 0.9, c = 0.95, we can be 95% sure, given the data, that the product is at least 90% reliable. Suppose that it is observed that the product succeeds all n times. Find a simple equation for c as a function of r.

I think this may be related to the beta or gamma distribution somehow but don’t really know how to approach.