Batches of 400 shells in the First World War were classified as 'accepted' or 'rejected' by testing a small number of shells from the batch. Tested shells are either 'good' or 'bad'; the probability that a randomly selected shell is good is p.

(a) In one testing method, eight shells from a batch (of 400) are selected at random and tested. The batch is accepted if at least three of these eight shells are good. Use a binomial distribution, with p=0.2, to find the probability that the batch is accepted.

(b) In a second testing method, each batch of 400 is subdivided into four sub-batches of 100 shells each. Two shells from each sub- batch are tested, and the sub-batch is accepted if at least one of the two shells is good. Use a binomial distribution, with P-0.2, (i) to show that the probability that one particular sub-batch is accepted is 0.36, (ii) to find the probability that, out of four sub-batches, at least three are accepted.