Recent results on the random probability of success of an experiment

The most popular hybrid frequentist-Bayesian method to evaluate the probability of success (PoS) of an experiment is based on the expected value of the power function of a test with respect to a prior distribution on the unknown parameter under scrutiny. Recently, it has been argued that exploring the whole distribution of the random power induced by a prior, instead of just its expected value, may provide a more accurate way to quantify the chance of success of the trial. In this article we review some of these results. Specifically, we consider relevant models where closed-form expressions of the cumulative distribution and density functions of the random power can be derived. After some considerations on the asymptotic behavior of the random power and PoS, we address the sample size determination problem.