Concrete is mixed according to statistically analyzed test results, on a plant-by-plant basis. Suppose that we grade a batch plant's output as either "A" good quality, "B" average quality or "C" poor quality, with respective probabilities of 0.25, 0.65 and 0.10. The probability of failure of a component from each of these qualities of mix would be 0.001, 0.01 and 0.1, respectively.

A) What would be the probability of failure of a component cast from a batch of concrete from this batch plant?

B) We can perform tests on the concrete mixes. Suppose that the probability of passing the test is 0.90, 0.70 and 0.25 respectively. If a batch passes the test:

1) What is the probability that the batch is of good quality?

2) If a batch passes the test, what is the probability of failure of a structural component cast from this batch of concrete?