A quality-control plan for the concrete in a nuclear reactor containment vessel calls for casting 6 cylinders for each batch of 10 3 poured and testing them as follows:

1 at 7 days

1 at 14 days

2 at 28 days

2 more at 28 days if any of first four cylinders is "inadequate"

The required strength is a function of age.

If the cylinder to be tested is chosen at random from those remaining (i. e., with equal likelihoods):

(a) What is the probability that all six will be tested if in fact one inadequate cylinder exists in the six?

(b) If the batch will be "rejected" if two or more inadequate cylinders are found, what is the likelihood that it will not be rejected given that exactly two are in fact inadequate? (Rejection will lead to more expensive coring and testing of concrete in place.)

(c). A "satisfactory" concrete batch gives rise to an inadequate cylinder with probability p = 0.1. (This value is consistent with present recommended practice.) What is the probability that there will be one or more inadequate cylinders in the six when the batch is "satisfactory"? (Assume independence of the quality of the individual cylinders.)

(d) Given that the batch is satisfactory (p = 0.1), what is the probability that the batch will be rejected? What is the probability that an unsatisfactory batch (in particular, say, p = 0.3) will not be rejected? Clearly a quality control plan wants to keep both these probabilities low, while also keeping the cost of testing small.