3.10 Guessing on an exam. In a multiple choice exam, there are 5 questions and 4 choices for each question \( (a, b, c, d) \). Nancy has not studied for the exam at all and decides to randomly guess the answers. What is the probability that: (a) the first question she gets right is the \( 3^{\text {rd }} \) question?

To find the probability that Nancy's first correct guess occurs on the 3rd question, we need to calculate the probability of her getting the first two questions wrong and then the third question right. Since there are 4 choices for each question, the probability of guessing incorrectly on a question is \( \frac{3}{4} \). For the first and second questions: - Probability of getting the first question wrong: \( \frac{3}{4} \) - Probability of getting the second question wrong: \( \frac{3}{4} \) For the third question: - Probability of getting the third question right: \( \frac{1}{4} \) Now we combine these probabilities. The total probability that the first correct answer is on the 3rd question is: \[ P(\text{First correct on 3rd}) = P(\text{1st wrong}) \times P(\text{2nd wrong}) \times P(\text{3rd correct}) = \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) \times \left(\frac{1}{4}\right) = \frac{9}{64}. \] So, the probability that Nancy’s first correct answer is the 3rd question is \( \frac{9}{64} \).