4. Machine 1 makes 30% of all parts in a factory, and Machine 2 makes the rest.

Question

4. Machine 1 makes 30% of all parts in a factory, and Machine 2 makes the rest. 2% of all parts made by Machine 1 are defective, and 3% of all parts made by Machine 2 are defective. (a) If a part is made by Machine 1, what is the probability it is defective? (b) What is the probability that a randomly chosen part will be defective and made by Machine 2? (c) What is the probability that a randomly chosen part will be non-defective and made by Machine 1? (d) What is the probability that a randomly chosen part is defective? (e) Suppose we find a part that is defective; what is the probability that it came from Machine 1?

Explanation / Answer

Assume : Total of 1000 parts

Machine 1 makes 30%, i.e 300 parts
Defective by machine 1 = 6 parts
Non-defective by machine 1 = 294 parts

Machine 2 makes 700 parts
Defective by machine 2 = 21 parts
Non-defective by machine 2 = 679 parts

a)

P(defective | machine 1) = 2% = 0.02

b)

P(defective and made by machine 2) = 21 parts / 1000 parts = 0.021

c)

P(non-def and made by machine 1) = 294/1000 = 0.294

d)

P(defective) = (6 + 21) / 1000 = 27/1000 = 0.027

e)

P(machine 1 given that it is defective) = 6/(6 + 21) = 6/27 = 2/9

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