I have this problem (from Montgomery's Applied Probability and Statistics, 5th Edition, problem 2-145, if anyone wants to see the original problem) but it's long, so for the sake of brevity I'll give a smaller version of the problem--if I can see how to solve the smaller version I'm sure I could figure out the bigger one.
Problem: Suppose a test claims 99% accuracy of detecting $A$ and 98% accuracy of detecting $B$, and 97% accuracy of detecting $C$. A collection of samples is formed, such that 40% of the samples contain $A$, 30% contain $B$, and 20% contain $C$. "The process signals if all categories $A$ through $C$ are present." Find the probability that the process signals, and then find the probability that $A$ is present given the process signaled.
Now that sounds like it must be a sloppy way of expressing the problem, because if the process signals then every category is present and therefore, conditional on the process signaling, $A$ is present with 100% probability. I'm guessing, instead, what should go in the place of the quoted sentence is "The process signals if all categories are detected by the test."
In that case it seems to me that we would find the probability that the test is positive for $A$, which is $P(A+) = P(A+|A)P(A)+P(A+|A')P(A')$, right? I think the accuracy statement tells me $P(A+|A)=0.99$, but how would I get $P(A+|A')$? Or am I just interpreting the whole question the wrong way?