Step 1 :The problem is asking for the probability that a car was both manufactured at plant A and has a defective emissions control device. This is a conditional probability problem.
Step 2 :We know that 50% of the cars are manufactured at plant A, and among those, 3% have defective emissions control devices.
Step 3 :So, the probability that a car was both manufactured at plant A and has a defective emissions control device is simply the product of these two probabilities.
Step 4 :Let's denote the probability of a car being manufactured at plant A as \(probability_A = 0.5\) and the probability of a car having a defective device given it was manufactured at plant A as \(probability_defective_given_A = 0.03\).
Step 5 :The probability that a car was both manufactured at plant A and has a defective emissions control device is \(probability_A\) times \(probability_defective_given_A\), which equals to \(0.5 * 0.03 = 0.015\).
Step 6 :Final Answer: The probability that one of this manufacturer's cars was manufactured at plant A and has a defective emissions control device is \(\boxed{0.015}\) or \(\boxed{1.5\%}\).