This paper develops Bayesian prediction intervals for the minimum of any specified number of future measurements from a Gumbel distribution based on previous observations. The need for such intervals arises in the analysis of data from outlet side feeder pipes at Ontario nuclear power plants. The issue is how to best use these measurements in order to arrive at a statistically sound conclusion concerning the minimum thickness of all remaining uninspected pipes, in particular with what confidence can it be asserted that the remaining wall thicknesses are above an acceptable minimum to ensure a sufficiently high thickness up to the end of the next operating interval. The result gives a probability measure of the potential benefit of performing additional inspections when considered against the additional radiation exposure and the cost of performing additional inspections. Previously, this problem was approached by adapting a classical prediction interval that was originally derived for normal data. Here we examine both a hybrid Bayesian method that combines Bayesian ideas with maximum likelihood and also a full Bayesian approach using Markov Chain Monte Carlo. We show that the latter gives larger lower prediction limits and therefore more margin to fitness for service.