In a factory that produces machine parts, 90% of the produced parts are good quality and 10% are of defective quality. However, a fault in the testing machine incorrectly identifies good-quality parts as defective 5% of the time and defective parts as good quality 2% of the time. What is the probability that a randomly selected part is defective, given that the testing machine identified it as defective?
0.65
0.84
0.96
0.89
Let A be the event that the part is defective, and B be the event that the testing machine identified the part as defective.
From the given information, we have:
P(A) = 0.10 (10% of the parts are defective)
P(B|A) = 0.95 (the machine correctly identifies 95% of the defective parts as defective)
P(B|A') = 0.02 (the machine incorrectly identifies 2% of the good parts as defective)
We want to find P(A|B), the probability that the part is defective given that the machine identified it as defective.
Using Bayes' theorem, we have:
We can calculate P(B) using the law of total probability:
Where P(A') = 1 - P(A) = 0.90 (90% of the parts are good quality)
Substituting the values, we get:
Substituting into Bayes' theorem, we get:
Therefore, the probability that a randomly selected part is defective given that the testing machine identified it as defective is 0.84.
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