If it helps, I think this is an example of a problem where they give different answers to the same problem. From Jaynes; see http://bayes.wustl.edu/etj/articles/confidence.pdf , page 22 for the details, and please let me know if I've erred or misinterpreted the example.

Three identical components. You run them through a reliability test and they fail at times 12, 14, and 16 hours. You know that these components fail in a particular way: they last at least X hours, then have a lifetime that you assess as an exponential distribution with an average of 1 hour. What is the shortest 90% confidence interval / probability interval for X, the time of guaranteed safe operation?

Frequentist 90% confidence interval: 12.1 hours - 13.8 hours

Bayesian 90% probability interval: 11.2 hours - 12.0 hours

Note: the frequentist interval has the strange property that we know for sure that the 90% confidence interval does not contain X (from the data we know that X <= 12). The Bayesian interval seems to match our common sense better.