A machine produces defective parts with two di erent probabi

A machine produces defective parts with two di erent probabilities depending on its state of
repair. If the machine is in good working order (G), it produces defective parts (D) with a 2%
chance. If it needs maintenance, it produces defective parts with a 30% chance. The probability that
the machine is in good working order is 80%

a)What is the probability that the machine produces defective parts? Draw an appropriate
tree diagram to answer this question

b)If it is known that the machine produced defective parts, what is the probability that the
machine needs maintenance?

Solution

a)

P(D) = P(G) P(D|G) + P(G\') P(D|G\')

= 0.80*0.02 + (1-0.80)*(0.30)

= 0.076 [ANSWER]

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b)

P(G\'|D) = P(G\') P(D|G)/P(D)

= (1-0.80)*(0.30)/0.076

= 0.789473684 [ANSWER]