Probability of Defective Components: Past Data Analysis & Quality Inspection

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SUMMARY

The probability of a component being declared defective is calculated using Bayes' theorem, considering the known defect rate of 0.5%. The quality inspection procedure identifies 90% of defective components and incorrectly flags 3% of non-defective components. The overall probability of a component being declared defective is 0.5% * 90% + 99.5% * 3%, resulting in approximately 3.485%. The conditional probability that a component is actually defective given that it is declared defective is calculated as (0.5% * 90%) / 3.485%, yielding about 14.3%.

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On the basis of past data, suppose it is known that 0.5% of a components of a particular type manufactured by a firm are defective. The quality inspection procedure is such that 90% of the defective components will actually be found defective and 3% of the non-defective components will also be wrongly declared defective. A component is selected at random from the manufacturing process and inspected.

a) What is the probability that the components is declared defective?

b) If the component is declared defective, what is the conditional probability that the item is actually defective?
 
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