Fasteners used in aircraft manufacturing are slightly crimped so that they lock enough to avoid loosening during vibration. Suppose that 95% of all fasteners pass an initial inspection. Of the 5% that fail, 20% are so seriously defective that they must be scrapped. The remaining fasteners are sent to a recrimping operation, where 40% cannot be salvaged and are discarded. The other 60% of these fasteners are corrected by the recrimping process and subsequently pass inspection.
a. What is the probability that a randomly selected incoming fastener will pass inspection either initially or after recrimping?
b. Given that a fastener passed inspection, what is the probability that it passed the initial inspection and did not need recrimping?