An automobile manufacturer produces 48% of its cars at plant A. If 6% of the cars manufactured at plant A have defective emissions control devices, what is the probability that one of this manufacturer’s cars was manufactured at plant A and has a defective emissions control device?
The correct answer and explanation is:
Correct Answer:
The probability that a car was manufactured at plant A and has a defective emissions control device is: P(A and D)=P(A)×P(D∣A)=0.48×0.06=0.0288P(A \text{ and } D) = P(A) \times P(D|A) = 0.48 \times 0.06 = \boxed{0.0288}
Explanation (Approx. 300 words):
This problem involves joint probability, which refers to the likelihood of two events occurring together. In this case:
- Event A: A car is manufactured at plant A.
- Event D: A car has a defective emissions control device.
We are given:
- P(A)=0.48P(A) = 0.48, meaning 48% of the cars are made at plant A.
- P(D∣A)=0.06P(D|A) = 0.06, meaning 6% of the cars made at plant A are defective in emissions control.
We are asked to find the probability that a car is both from plant A and has a defective emissions control device. This is the joint probability of both events occurring. The formula for joint probability when one event is conditional on another is: P(A and D)=P(A)×P(D∣A)P(A \text{ and } D) = P(A) \times P(D|A)
Here, we multiply the probability that a car comes from plant A (48%) by the probability that a car from plant A is defective (6%): P(A and D)=0.48×0.06=0.0288P(A \text{ and } D) = 0.48 \times 0.06 = 0.0288
This means there is a 2.88% chance that a randomly selected car from this manufacturer is both produced at plant A and has a defective emissions control device.
This approach assumes the data is representative and that the events are not influenced by other factors (e.g., quality control varying over time). Understanding joint probability is important in quality control, risk assessment, and decision-making processes across manufacturing and operations.
So, the final answer is: 0.0288 or 2.88%\boxed{0.0288} \text{ or } \boxed{2.88\%}