To solve this problem, we need to use Bayes' Theorem. Let's break down the information given: 1. **P(Correct | Guessed)** = \(\frac{1}{2}\): The probability of answering correctly if the answer is guessed. 2. **P(Guessed | Correct)** = \(\frac{1}{6}\): The probability that the answer was guessed given that it was correct. We want to find the probability that the student knows the answer to a randomly chosen question, i.e., **P(Knows)**. ### Using Bayes' Theorem Bayes' theorem states: \[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \] Here, we can set: - **A** as the event "Knows" - **B** as the event "Correct" ### Applying Bayes' Theorem From the given information: \[ P(\text{Correct | Guessed}) = \frac{1}{2} \] \[ P(\text{Guessed | Correct}) = \frac{1}{6} \] Let's assume: - \( P(\text{Knows}) = x \) - \( P(\text{Guessed}) = 1 - x \) We know: \[ P(\text{Correct}) = P(\text{Correct | Knows}) \cdot P(\text{Knows}) + P(\text{Correct | Guessed}) \cdot P(\text{Guessed}) \] Given that \( P(\text{Correct | Knows}) = 1 \), we have: \[ P(\text{Correct}) = 1 \cdot x + \frac{1}{2} \cdot (1 - x) \] \[ P(\text{Correct}) = x + \frac{1}{2} - \frac{x}{2} \] \[ P(\text{Correct}) = \frac{x}{2} + \frac{1}{2} \] Using Bayes' theorem: \[ P(\text{Guessed | Correct}) = \frac{P(\text{Correct | Guessed}) \cdot P(\text{Guessed})}{P(\text{Correct})} \] \[ \frac{1}{6} = \frac{\frac{1}{2} \cdot (1 - x)}{\frac{x}{2} + \frac{1}{2}} \] Solving for \( x \): \[ \frac{1}{6} = \frac{\frac{1}{2} - \frac{x}{2}}{\frac{x}{2} + \frac{1}{2}} \] Cross-multiplying gives: \[ \frac{1}{6} \left(\frac{x}{2} + \frac{1}{2}\right) = \frac{1}{2} - \frac{x}{2} \] \[ \frac{x}{12} + \frac{1}{12} = \frac{1}{2} - \frac{x}{2} \] Multiplying through by 12: \[ x + 1 = 6 - 6x \] \[ 7x = 5 \] \[ x = \frac{5}{7} \] So, the probability that the student knows the answer to a randomly chosen question is \(\frac{5}{7}\).