Fasteners used in aircraft manufacturing are slightly...

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?

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?

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Key Concepts

Law of Total Probability

This principle involves breaking down a complex event into a set of mutually exclusive scenarios and summing their individual probabilities. In quality control and manufacturing contexts, such as the inspection and rework of fasteners, it is used to combine the probability of passing through multiple pathways (e.g., passing on the first attempt or after reprocessing) to calculate an overall success or failure rate.

Conditional Probability

Conditional probability is the concept of determining the likelihood of an event given that another event has occurred. In the context of inspection processes, it helps to determine the chances that a fastener passed the initial inspection given that it eventually passed overall inspection, thereby providing insights into the effectiveness of first-line quality control measures.

Yield Analysis in Multi-Stage Processes

This concept focuses on the cumulative yield of products that go through multiple sequential processing or inspection stages. In manufacturing, each stage—such as initial inspection and subsequent rework—contributes to the final yield. Understanding the yield at each stage and how they combine is crucial for optimizing overall production efficiency and quality.

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