Question

A stationary store has decided to accept a large shipment of ballpoint pens if an inspection of 20 randomly selected pends yields no more than two defective pens. (a) Find the probability that this shipment is accepted if 5% of the total shipment is defective. (b) Find the probability that this shipment is not accepted if 15% of the total shipment is defective. Kindly use the numbers given in the word sentences to show the work. Thank you

Answer 1

a) n = 20

   p = 0.05

It is a binomial distribution.

P(X = x) = nCx * p^x * (1 - p)^{n - x}

P(X < 2) = P(X = 0) + P(X = 1) + P(X = 2)

             = 20C0 * (0.05)^0 * (0.95)^20 + 20C1 * (0.05)^1 * (0.95)^19 + 20C2 * (0.05)^2 * (0.95)^18

             = 0.9245

The probability that the shipment is accepted = 0.9245

b) n = 20

   p = 0.15

P(X > 2) = 1 - P(X < 2)

            = 1 - (P(X = 0) + P(X = 1) + P(X = 2))

            = 1 - (20C0 * (0.15)^0 * (0.85)^20 + 20C1 * (0.15)^1 * (0.85)^19 + 20C2 * (0.15)^2 * (0.85)^18)

            = 1 - 0.4049

            = 0.5951

The probability that the shipment is not accepted = 0.5951