An inspector working for a manufacturing company has a 95 ch

An inspector working for a manufacturing company has a 95% chance of correctly identifying defective
items and 2% chance of incorrectly classifying a good item as defective. The company has evidence that its line
produces 1% of nonconforming items.

(a) What is the probability that an item selected for inspection is classified as defective?
(b) If an item selected at random is classified as nondefective, what is the probability that it is indeed good?

Show Step by step please, Thank You

Solution

a)

P(def) = P(good) P(def|good) + P(bad) P(def|bad) = (1-0.99)*0.02 + 0.01*0.95

= 0.0293 [answer]

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b)

As

P(nondef) = 1 - P(def) = 0.9707

And

P(good|nondef) = P(nondef and good) / P(nondef)

As

P(nondef and good) = P(good) P(nondef|good) = (1-0.01)*0.95 = 0.9405

Then

P(good|nondef) = P(nondef and good) / P(nondef) = 0.9405/0.9707 = 0.968888431 [answer]