To solve the problem of finding the probability that a student will not pass any of the three exams, we need to follow a systematic approach. Let's break it down step-by-step. ### Step 1: Understand the Given Information - The probability of passing any single exam is given as \( P(\text{pass}) = \frac{2}{3} \). - Consequently, the probability of failing a single exam can be calculated as: \[ P(\text{fail}) = 1 - P(\text{pass}) = 1 - \frac{2}{3} = \frac{1}{3} \] ### Step 2: Calculate the Probability of Failing All Three Exams Since the exams are independent events (the outcome of one exam does not affect the others), we can find the probability of failing all three exams by multiplying the probability of failing each individual exam: \[ P(\text{fail all three}) = P(\text{fail}) \times P(\text{fail}) \times P(\text{fail}) = P(\text{fail})^3 \] Substituting the value we found for \( P(\text{fail}) \): \[ P(\text{fail all three}) = \left(\frac{1}{3}\right)^3 = \frac{1}{27} \] ### Step 3: Identify the Correct Option Now that we have calculated the probability of the student failing all three exams, we can compare it with the provided options: - A. \( \frac{2}{3} \) - B. \( \frac{4}{9} \) - C. \( \frac{8}{27} \) - D. \( \frac{1}{27} \) The correct answer is **D. \( \frac{1}{27} \)**. ### Step 4: Explain Why Other Options Are Incorrect - **Option A: \( \frac{2}{3} \)** - This option represents the probability of passing a single exam, not the probability of failing all three. - **Option B: \( \frac{4}{9} \)** - This value does not correspond to any relevant calculation regarding passing or failing multiple exams. It could be a miscalculation of probabilities. - **Option C: \( \frac{8}{27} \)** - This value might represent the probability of passing two exams and failing one, but it does not apply to the scenario of failing all three exams. ### Step 5: Summary of Key Concepts - The probability of failing an exam is the complement of passing. - For independent events, the total probability of multiple events occurring is the product of their individual probabilities. - Always ensure to check the independence of events when calculating combined probabilities. ### Revision Summary - The probability of failing a single exam is \( \frac{1}{3} \). - The probability of failing all three exams is \( \left(\frac{1}{3}\right)^3 = \frac{1}{27} \). - The correct answer is **D. \( \frac{1}{27} \)**. - Understanding the independence of events is crucial for calculating combined probabilities correctly.