2021 Jan Question 2 In a batch of manufactured items, the proportion of defective items is p. From each batch a random sample of nine is taken and tested. If two or more items are found to be defective, the batch is rejected, otherwise it is accepted. (a) Show that the probability that a batch is accepted is (1 - p)^8 (1 +8p). (b) It is decided to modify the checking scheme such that when one defective is found in the sample, a second sample of nine is taken and the batch rejected if this contains any defectives. With this exception, the original scheme is continued. (i) Apply the modified scheme and derive an expression in terms of p for the probability that a batch is accepted, simplifying your answer as much as possible. (ii) Evaluate and comment on the average number of items sampled over a large number of batches, with this new sampling scheme, when p = 0.1. Provide your answer to 2 decimal places.

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2021 jan question 2 in a batch of manufactured items the proportion of

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2021 Jan Question 2 In a batch of manufactured items, the proportion of defective items is p. From each batch a random sample of nine is taken and tested. If two or more items are found to be defective, the batch is rejected, otherwise it is accepted. (a) Show that the probability that a batch is accepted is (1 - p)^8