Defective Subpopulation Distribution and Conditional Probability A colleague and I have tried two different approaches to this problem, both of which seem to make sense but are resulting in very different answers.
Suppose we have some units undergoing B hours of testing and are interested in the probability of failure in the next hour (or a generic next time t) given successful completion of B. We have some data showing that about 15% of the units failed in the B hours of testing at various times, and have tried fitting various models to estimate the answer.
A favored approach by both of us is to use a defective subpopulation distribution: We estimate the distribution parameters (call the distribution CDF) and the proportion defective p; no problem there. The problem is how to use the model to estimate the “next time t” probability of failure.
My approach is based on conditional reliability: R(t|survived B)=R(B+t)/R(B). I set R(t)=1-p*CDF(t), plug into the conditional reliability equation, and take one minus the answer to convert back to probability of failure.
My colleague takes a different approach: He says the probability of failure is equal to p*Prob(failure in next hour|member of defective subpopulation)=p(CDF(B+t)-CDF(B))/(1-CDF(B))
Both approaches seem to make sense but the answers are off by about an order of magnitude. I of course think mine is correct but can’t clearly explain why his isn’t. Any ideas?

After thinking through this a bit more I was able to confirm that my approach is correct and identify where my colleague's approach went wrong: His use of 'p' (the probability of being in the defective subpopulation) in p*Prob(failure in next hour|member of defective subpopulation) should be replaced with Prob(member of defective subpopulation | survived B).
That probability can be computed with a fairly straightforward application of Bayes' rule and a bit of algebra (or a calculator) then shows that our answers are the same.
