Question

5. Seventy percent of all vehicles examined at a certain emissions inspection station pass the inspection. Assuming that successive vehicles pass or fail independently of one another, calculate the following probabilities: a. P(all of the next three vehicles inspected pass) b. P(at least one of the next three inspected fails) c. P(exactly one of the next three inspected passes) d. Pat most one of the next three vehicles inspected passes) e. Given that at least one of the next three vehicles passes inspection, what is the probability that all three pass (a conditional probability)?

Answer 1

P(pass the inspection) = 0.7

P(fail the inspection) = 1 - 0.7 = 0.3

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

5)a) P(X = 3) = 3C3 * (0.7)^3 * (0.3)^0 = 0.343

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

                  = 1 - P(X = 0)

                 = 1 - (3C0 * (0.3)^0 * (0.7)^3)

                 = 1 - 0.343 = 0.657

c) P(X = 1) = 3C1 * (0.7)^1 * (0.3)^2 = 0.189

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

                  = 3C0 * (0.7)^0 * (0.3)^3 + 3C1 * (0.7)^1 * (0.3)^2

                  = 0.216

e) P((X = 3) | (X > 1)) = P((X = 3) and (X > 1))/P(X > 1)

                                   = P(X = 3)/P(X > 1)

                                   = (3C3 * (0.7)^3 * (0.3)^0)/(1 - (3C0 * (0.7)^0 * (0.3)^3))

                                   = 0.343/0.973

                                   = 0.3525