In a bolt factory, there are four machines m1, m2, m3, and m4, producing 20%, 15%, 25%, and 40% of the total output respectively. The fractions of defective bolts produced by these machines are respectively 0.05, 0.04, 0.03, and 0.02. A bolt is chosen at random from the factory's production. What is the probability that it is defective? Given that it is defective, what is the probability that it was produced by machine m2
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Probability that the defected bolt is from m2 = 4/21
Probability that the bolt is produce from m1 = P(m1) = 0.2
Probability that the bolt is produce from m2 = P(m2) = 0.15
Probability that the bolt is produce from m3 = P(m3) = 0.25
Probability that the bolt is produce from m4 = P(m4) = 0.4
The fractions of defective bolts produced by these machines are respectively 0.05, 0.04, 0.03, and 0.02.
Let the Probability that the defective bolt is from m2 be x.
Using Bayes rule -
x = (P(m2) × 0.04) / ( (P(m1) × 0.05) (P(m2) × 0.04) (P(m3) × 0.03) (P(m4) × 0.02 ) )
x = (0.15 × 0.04) / ( (0.2 × 0.05) (0.15 × 0.04) (0.25 × 0.03) (0.4 × 0.02 ) )
x = 4/21