From a very large batch of mass produced widgets, a simple random sample of 20 is selected and each widget in the sample is inspected to see whether or not it meets the quality requirement. The batch is accepted if the sample contains 0 or 1 defective widgets. In a simple random sample, every widget has the same probability of being selected for the sample. Calculate the exact probability of accepting a batch if it contains a proportion p of defectives, for p = 0.01, 0.05, 0.1.


(ii) The above scheme is now modified in that, if the sample contains more than 2 defectives the batch is still rejected, but if the sample of 20 contains just 2 defectives, a further random sample of size 20 is selected, which may be assumed independent of the first sample. If the second sample contains no defectives then the batch is accepted, otherwise the batch is rejected. List the possible numbers of defectives in the two samples which lead to acceptance of the batch, and hence calculate the probability of accepting a batch, for p = 0.01, 0.05, 0.1.


(iii) Rejected batches are subject to 100% inspection and all defectives are removed. If the batch size is 1000, calculate the expected total number of items inspected in the two sampling schemes in (i) and (ii), for each of the values of p given. Comment briefly on your results.