Question

A machine produces parts that are either defect free (90%), slightly defective (3%), or obviously defective. Prior to shipment produced parts are passes through an automatic inspection machine, which is supposed to be able to detect any part that is obviously defective and discard it. However, the inspection machine is not perfect. A part is incorrectly identified as defective and discarded 2% of the time that a defect free part is input. Slightly defective parts are marked defective and discarded 40% of the time, and obviously defective parts are correctly identified 98% of the time.

a. What is the total probability that a part is marked as defective and discarded by the automatic inspection machine?

b. What is the probability that a part is 'good' (either defect free or slightly defective) given that it makes it through the inspection machine and gets shipped?

c. What is the probability that a part is 'bad' (obviously defective) given that it makes it through the inspection machine and gets shipped?  

Answer 1

Pr(Defect free) = 0.90

Pr(Slightly Defective) = 0.03

Pr(Obviously defective) = 1 - 0.03 - 0.90 = 0.07

Pr(Defective l when not defective) = 0.02

Pe(Defective l slightly defective parts) = 0.4

Pr(Defective parts correcty identified as defective) = 0.98

(A) Pr(Marked as defective and discarded by the machine) = Pr(Not defective) * Pr(Discarded it) + Pr(Slightly Defective) * Pr(Discarded it) + Pr(Defective) * Pr( Discarded it)

= 0.90 * 0.02 + 0.03 * 0.4 + 0.07 * 0.98 = 0.0986

(B) Pr(Good parth that is shipped) = 1 - Pr(Not discarded by machine) = 1 - 0.0986 = 0.9014

Pr(Good part that is shipped) = Pr(Defect free) * Pr(Not discarded) + Pr(Slightly defective) * Pr(Not discarded)

= 0.90 * 0.98 + 0.03 * 0.60 = 0.9

Pr(A part is good when shipped) = 0.9/0.9014 = 0.9984

(c) Pr(Bad part get shipped) = 1 - 0.9984 = 0.0016