I interpret the question along the lines of "if they do a hundred launches, what's your probability distribution on the number of them that fail?"
If I have a normal coin and a coin which is double-sided (but I don't know which side), I'll give 50% for both of them coming up heads next time I toss. But if I toss them a hundred times each, my probability distributions for them look totally different.
I suspect this information is encoded in your priors, but I don't know offhand how to access it.
So thinking about this some more: we're assuming engines have some "true failure rate" which we're trying to divine from evidence. 2% is currently the mean of your distribution on TFR, but the relevant question is what's the variance of your distribution?
Uniform prior, updated on the evidence that 1/36 engines have failed, I think this gives P(TFR = x) = x(1-x)^35 / (int x(1-x)^35 dx from 0 to 1). Apparently the integral is 1/1332, so P(TFR=x) = 1332·x·(1-x)^35. But that seems to have a mean of 5%, compared to your value of 2%, so I may have done something dumb?
You can answer that question for each prior. The probability of, say, 3 failures is the sum over all priors of the probability that that prior is true, times the probability that it would leave you with 3 failures.
I leave writing a program to calculate this as an exercise to the reader. I've put enough time in on this one already, I have paying work to get back to.
If I have a normal coin and a coin which is double-sided (but I don't know which side), I'll give 50% for both of them coming up heads next time I toss. But if I toss them a hundred times each, my probability distributions for them look totally different.
I suspect this information is encoded in your priors, but I don't know offhand how to access it.