Ninety percent of all vehicles examined at a certain emissions inspection statio

Question

Ninety 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. (Enter your answers to three decimal places.)

(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)    P(at 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)? (Round your answer to three decimal places.)

Explanation / Answer

P(the vehicle passes the inspection) = 0.90

P(the vehicle does not pass the inspection) = 1 - 0.90 = 0.10

a)

P(all of the next three vehicles inspected pass) = P(1st vehicle passes inspection).(2nd vehicle passes inspection).(3rd vehicle passes inspection)

= (0.90)(0.90)(0.90) = 0.729

b)

P(at least one of the next three inspected fails) = 1 - P(none of the next three inspected fails)

= 1 - (0.90)(0.90)(0.90) = 0.271

c)

P(exactly one of the next three inspected passes) = (0.90)(0.10)(0.10) + (0.10)(0.90)(0.10) + (0.10)(0.10)(0.90)

= 0.027

d)

P(at most one of the next three vehicles inspected passes) = P(none of the next three inspected vehicles pass) + P(exactly one of the next three inspected vehicles pass)

= (0.10)(0.10)(0.10) + (0.90)(0.10)(0.10) + (0.10)(0.90)(0.10) + (0.10)(0.10)(0.90) = 0.028

e)

Required probability = P(all of the next three vehicles inspected pass given that at least one of the next 3 vehicles passes the inspection)

= P(all of the next three vehicles inspected pass and that at least one of the next 3 vehicles passes the inspection) / P(at least one of the next 3 vehicles passes the inspection)

= P(all of the next three vehicles inspected pass the inspection) / P(at least one of the next 3 vehicles passes the inspection)

= [(0.90)(0.90)(0.90)] / [1 - (0.10)(0.10)(0.10)]

= 27 / 37

= 0.7297

Related Questions

Navigate

• Browse (All)
• Browse N
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.