Suppose that a machine shop orders 500 bolts from a supplier. To determine whether to accept the shipment of bolts, the manager of the facility randomly selects 12 bolts. If none of the 12 randomly selected bolts is found to be defective, he concludes that the shipment is acceptable.

(a) If 10% of the bolts in the population are defective, what is the probability that none of the selected bolts are defective?

(b) If 20% of the bolts in the population are defective, what is the probability that none of the selected bolts are defective?

Approach This is a hypergeometric experiment with N = 500 and n = 12. In part (a), we have that k = 0.115002 = 50 defectives. The possible values of the random variable X are x = 0, 1, 2,c, 12 (you cannot have more successes than the sample size). Notice that a success means finding a defective bolt. In part (b), we have that k = 0.215002 = 100 defectives. The possible values of the random variable X are x = 0, 1, 2,c, 12. We use Formula (1) to compute the probabilities.

Formula 1

(C) (N-KCn-x) P(x) (1) NCn