We receive a shipment from one of three warehouses W1, W2, and W3, where each wa

Question

We receive a shipment from one of three warehouses W1, W2, and W3, where each warehouse is equally likely and: • Warehouse 1 has 50 parts, 5 of which are defective. • Warehouse 2 has 40 parts, 10 of which are defective. • Warehouse 3 has 20 parts, 10 of which are defective.

(a) A shipment of 10 parts arrives and I test the first part. What is the probability that part is defective?

(b) A shipment of 10 parts arrives and I test all 10 parts. What is the probability that exactly 8 of the parts are defective?

(c) A shipment of 10 parts arrives and I test all 10 parts. I find exactly 8 defective parts.

• What is the probability the shipment came from warehouse W1?

• What is the probability the shipment came from warehouse W2?

• What is the probability the shipment came from warehouse W3?

(d) A shipment of 10 parts arrives and I test the first part, finding that it is defective. Given this information, what is the probability that I find exactly 8 defective parts (including the one already tested) in that shipment?

Explanation / Answer

A) P( defective 1st part) = (1/3)x(5/50) + (1/3)x(10/40) + (1/3)x(10/20)

= 0.2833

B) P(8 defective out of 10) = 0 + P(8 defective from W2) +

P(8 defective from W2)

= 0 + 10C8 x 0.258 + 0.752 + 10C8 x 0.58x0.52

0.0004 + 0.0439= 0.0443

C) P(shipment came from W1) = 0

P( shipment came from W2) = 0.009

P(shipment came from W3) = 0.991

D) P(8 parts defective | first part defective)

= P(8parts defective and 1st part defective)/P(first part defective)

= (0+ 0.25 x 9C7x0.257x0.752 + 0.5x9C7x0.57x0.52)/0.2833

= 0.125

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