Question: An automobile manufacturer produces 62% of its cars at plant A. If 5% of the cars manufactured at…

An automobile manufacturer produces 62% of its cars at plant A. If 5% 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?

Probability = (Type an integer or decimal.)

Solution

The problem asks for the probability that a car was manufactured at plant A and has a defective emissions control device. First, let’s define the events: - Let \( A \) be the event that a car is manufactured at plant A. - Let \( D \) be the event that a car has a defective emissions control device. The probability that a car is manufactured at plant A is given by: \[ P(A) = 0.62 \] The probability that a car has a defective emissions control device given that it was manufactured at plant A is: \[ P(D \mid A) = 0.05 \] We need to find the probability that a car was manufactured at plant A and has a defective emissions control device, which is \( P(A \cap D) \). Using the multiplication rule for probabilities, we have: \[ P(A \cap D) = P(A) \times P(D \mid A) \] Substituting the given probabilities: \[ P(A \cap D) = 0.62 \times 0.05 \] Now, calculate the product: \[ P(A \cap D) = 0.031 \] Thus, the probability that one of the manufacturer’s cars was manufactured at plant A and has a defective emissions control device is 0.031.