SET UP: You manufacture silicon chips. You have performed very expensive tests t

Question

SET UP: You manufacture silicon chips. You have performed very expensive tests to determine the following: the probability of manufacturing a defective chip is 4/10 the probability of passing inspection (by human inspectors) given that the chip is defective is 3/10 the probability of passing inspection (by human inspectors) given that the chip is non-defective is 8/10. MAIN QUESTION: What is the probability that a chip is defective given that it passed inspection (by human inspectors)? ANALYTICAL SOLUTION: Define the following events: Dthe event that a chip is defective D the event that a chip is non-defective G = the event that a chip passes inspection . G the event that a chip does not pass inspection (Fill up the blanks) P(D) = P(Dc) P(GID) PlGID",-

Explanation / Answer

Given: P(D) =0.4

P(G|D) = 0.3

P(G|Dc) = 0.8

Thus, P(Dc) = 1 - P(D) =0.6

P(Gc|D) = 1 - P(G|D) = 0.7

P(Gc|Dc) = 1 - P(G|Dc) = 0.2

Now, MAIN Question

P(D|G) = P(G|D)*P(D)/P(G)

Now, P(G) = P(G|D)*P(D) + P(G|Dc)*P(Dc) = 0.3*0.4 + 0.8*0.6= 0.6

Thus, P(D|G) = 0.3*0.4/0.60 = 0.2

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