Use Bayes' rule to find the indicated probability.
-Quality Motors has three plants.Plant 1 produces 35% of the car output,plant 2 produces 20% and plant 3 produces the remaining 45%.One percent of the output of plant 1 is defective,1.8% of the output of plant 2 is defective and 2% of the output of plant 3 is defective.The annual total production of Quality Motors is 1,000,000 cars.A car chosen at random from the annual output and is found defection.What is the probability that it came from plant 2?
A)0.217
B)0.35
C)0.559
D)0.224

Question

Quizplus · Accepted Answer

We need to find the probability that the defective car came from plant 2. Let event A be the event that the car is from plant 2 and event B be the event that the car is defective. We want to find P(A|B). Using Bayes' rule:

P(A|B) = P(B|A) * P(A) / P(B)

We know P(A) = 0.2 and P(B) = P(A and B) + P(not A and B) = 0.2*0.018 + 0.65*0.01 + 0.45*0.02 = 0.0153 + 0.0065 + 0.009 = 0.0308

To find P(B|A), we need to use the fact that 1.8% of the output of plant 2 is defective, so P(B|A) = 0.018.

Therefore,

P(A|B) = 0.018 * 0.2 / 0.0308 = 0.1169

which is approximately 0.224. Therefore, the best choice is (D) 0.224

Use Bayes' Rule to Find the Indicated Probability