Question

(b) whats the probability that the part is defective? (round to 2 decimal places. leave your answer in decimal form.) 0.21 (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.) 0.286

Explanation:

Step1: Calculate the probability that the part is defective

Use the law of total probability. Let \(P(A) = 0.2\), \(P(B)=0.3\), \(P(C) = 0.5\) (probabilities of parts being produced by machines A, B, C respectively). Let \(P(D|A)=0.01\), \(P(D|B)=0.02\), \(P(D|C)=0.03\) (probabilities of a part being defective given it is produced by machine A, B, C respectively).

The law of total probability formula is \(P(D)=P(A)P(D|A)+P(B)P(D|B)+P(C)P(D|C)\)

Substitute the values:

$$
LATEXBLOCK0
$$

Step2: Calculate the probability that the defective part was produced by machine C

Use Bayes' theorem. Bayes' theorem formula is \(P(C|D)=\frac{P(C)P(D|C)}{P(D)}\)

We know \(P(C) = 0.5\), \(P(D|C)=0.03\) and \(P(D) = 0.023\)

Substitute the values:

$$
LATEXBLOCK1
$$

Answer:

(b) \(0.02\) (rounded to 2 decimal places)

(c) \(0.652\) (rounded to 3 decimal places)