. In some manufacturing environments, all manufactured items are inspected, and when the first defect is encountered, the operation is halted for investigation of possible causes. Suppose that the rate of defective items is 1% and that the quality of each item is independent of the quality of all other items.We are interested in how long we must wait to get the first defective item.
(a) The probability of the first inspected item being defective is 0.01. What is the probability that the first item is not defective
and the second is defective?
(b) What is the probability that the first two items are not defective and the third item is defective? This is the probability that
the first defective item is the third item inspected.
(c) Now you see the pattern. What is the probability that the first defective item is the fourth item inspected? The fifth item? Give
the general result: what is the probability that the first defective item is the kth item inspected? (Comment: The distribution of the number of trials to the first success is called a geometric distribution. In this exercise you have found geometric distribution probabilities when the probability of a success on each trial is p = 0.01. The same idea works for any p.)