A manufacturer submits a large batch of items per week to a consumer. To monitor quality control on each batch, the consumer adopts a two-stage sampling plan for decision making. In the first stage, a sample of 10 items is selected from the batch. The batch is accepted if this sample contains no defective items, and is rejected if the sample contains two or more defective items. Otherwise, a second sample of 10 items is selected. The batch is accepted if this second sample contains zero or one defective item or rejected otherwise. Suppose the proportion of defective items in the batch is 0.08.
a) Find the probability that a given batch will be rejected.
b) What is the average number of items inspected per batch?
c) The consumer will disqualify the manufacturer if, within any five consecutive batches, two are rejected. Find the probability that the manufacturer is disqualified within the first six weeks.