A quality control inspector is inspecting newly produced items for faults. The inspector searches an item for faults in a series of independent xations, each of a xed duration. Given that a aw is actually present, let p denote the probability that the aw is detected during any one xation (this model is dis- cussed in Human Performance in Sampling In- spection, Hum. Factors, 1979: 99-105).

a. Assuming that an item has a aw, what is the probability that it is detected by the end of the second  xation (once a  aw has been detected, the sequence of xations terminates)?

b. Give an expression for the probability that a aw will be detected by the end of the nth xation.

c. If when a aw has not been detected in three xa- tions, the item is passed, what is the probability that a awed item will pass inspection?

d. Suppose 10% of all items contain a aw [P(ran- domly chosen item is awed) = .1]. With the as- sumption of part (c), what is the probability that a randomly chosen item will pass inspection (it will automatically pass if it is not awed, but could also pass if it is awed)?

e. Given that an item has passed inspection (no aws in three xations), what is the probability that it is actually awed? Calculate for p = .5.