A quick quiz consists of a multiple-choice question with 3 possible answers followed by a multiplechoice question with 5 possible answers. If both questions are answered with random guesses, find the probability that both responses are correct. Give your answer as a percent rounded to one decimal place accuracy. You need not enter the "\%" symbol. $\%$
Solution
Step 1 ：A quick quiz consists of a multiple-choice question with 3 possible answers followed by a multiple-choice question with 5 possible answers. If both questions are answered with random guesses, we need to find the probability that both responses are correct.
Step 2 ：The probability of guessing the correct answer for the first question is 1 out of 3, which is approximately \(0.3333333333333333\).
Step 3 ：The probability of guessing the correct answer for the second question is 1 out of 5, which is \(0.2\).
Step 4 ：Since these are independent events, the probability of both events occurring is the product of their individual probabilities. So, we multiply \(0.3333333333333333\) and \(0.2\) to get \(0.06666666666666667\).
Step 5 ：Converting this to a percentage gives us approximately \(6.666666666666667\) percent.
Step 6 ：Final Answer: The probability that both responses are correct is approximately \(\boxed{6.7\%}\).
