Question

You are employed by a firm that manufactures computer chips. This firm receives components from two different suppliers. Currently, 65% of the components come from Supplier 1 and 35% of the components come from Supplier 2. Define A1 as the probability that a randomly-selected part comes from Supplier 1 and A2 as the probability that a randomly selected part comes from Supplier 2. In other words, Pr(A1) = .65 and Pr(A2) = .35.

Parts can be either good (G) or bad (B). We know from past experience that the probability that a part is bad varies based on the supplier it came from. Conditional on coming from Supplier 1, the probability that a part is good is .98, i.e. Pr(G | A1) = .98. The probability that a part is good, conditional on that part coming from Supplier 2 is .95, i.e. Pr(G | A2) = .95.

a.) What is the probability that any given part, chosen at random, is bad?

b.) What is the probability that of 5 chosen parts, at least one of the 5 parts will be bad?

c.) Suppose that you drew a part at random and discovered that it was bad. Your boss wants to know which supplier provided you the bad part. Calculate the probability that this bad part came from Supplier 1. Then calculate the probability that this bad part came from Supplier 2. Which supplier is more likely to have provided the bad part?

Answer 1

Pr(A1) = .65 and Pr(A2) = .35

Pr(G | A1) = .98

Pr(G | A2) = .95

a)

P(G) = P(A1) * P(G| A1) + P(A2) *P(G| A2) =                                    0.65*0.98+0.35*0.95=               0.9695

P(bad) = 1 - P(G) = 1 - 0.9695 = 0.0305

b)

n=5

p = 0.0305

P(at least one of the 5 parts will be bad) = 1 - P(no bad part) = 1 - (1-0.0305)^5 = 1-0.8565 = 0.1435

c)

P(A1| B) = P(A1)*P(B| A1)/P(B)=               0.65*(1-0.98) /0.0305=       0.4262

P(A2| B) = P(A2)*P(B| A2)/P(B)=               0.35*(1-0.95)/0.0305=       0.5738

supplier A2 is more likely to have provided the bad part