Question

three machines - a, b, and c - are used to produce a large quantity of identical parts at a factory. machine a produces 60% of the parts, while machines b and c produce 30% and 10% of the parts, respectively. historical records indicate that 10% of the parts produced by machine a are defective, compared with 30% for machine b and 60% for machine c. suppose we randomly select a part produced by one of these three machines at random.
(a) draw a tree diagram to model this chance process. complete the tree diagram by dragging the probabilities and placing them in the boxes.
(b) whats the probability that the part is defective?
(round to 2 decimal places. leave your answer in decimal form.)
(c) if the part is inspected and found to be defective, whats the probability it was produced by machine c?
(round to 3 decimal places. leave your answer in decimal form.)

Explanation:

Step1: Calculate the probability that the part is defective

Let \(P(A) = 0.6\), \(P(B)=0.3\), \(P(C) = 0.1\)
Let \(P(D|A)=0.1\), \(P(D|B) = 0.3\), \(P(D|C)=0.6\)
Using the law of total - probability \(P(D)=P(A)P(D|A)+P(B)P(D|B)+P(C)P(D|C)\)

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Step2: Calculate the probability that the defective part is produced by Machine C

Using Bayes' theorem \(P(C|D)=\frac{P(C)P(D|C)}{P(D)}\)
We know \(P(C) = 0.1\), \(P(D|C)=0.6\) and \(P(D) = 0.21\)

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Answer:

(b) \(0.21\)
(c) \(0.286\)