The easiest problem to fix is, you didn't specify any priors.
Read the whole thing more carefully. I started with the calculation for the maximum likelihood estimate, but I ended with a prior with equal a priori likelihoods for failure rates 0%, 0.01%, 0.02%, ..., 99.99%, 100%. This is a reasonable discretization of uniform on the unit interval.
My reply might be a little smarty-pants. Sorry if it is!
I would suggest that before indulging a tendency to be a smarty pants, that it is good to read the whole thing.
You're right, the second half of your post in effect puts a two-atom prior on "p", with zero everywhere else, and then goes on to use a sequence of such atoms, which would approximate a uniform prior. It's more standard to use smooth priors, because we don't have precise information, but you are right, I was not reading carefully.
You are still in error that there is not a way to describe the uncertainty in your estimate of the posterior probability of system failure. It has a posterior distribution, like everything else in a Bayesian analysis. You would compute it as I described -- Monte Carlo would be easiest.
Read the whole thing more carefully. I started with the calculation for the maximum likelihood estimate, but I ended with a prior with equal a priori likelihoods for failure rates 0%, 0.01%, 0.02%, ..., 99.99%, 100%. This is a reasonable discretization of uniform on the unit interval.
My reply might be a little smarty-pants. Sorry if it is!
I would suggest that before indulging a tendency to be a smarty pants, that it is good to read the whole thing.