In a test, an examinee either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he makes a guess is `1/3` and the probability that he copies the answer is 1/6. The probability that his answer is correct, given that he copied it, is 1/8. Find the probability that he knew the answer to the question, given that be correctly answered it.

Options

A. a. `17/39`
B. b. `13/29`
C. c. `24/29`
D. d. `24/39`

Answer

To solve the problem step by step, we will use the concepts of probability and Bayes' theorem.

### Step 1: Define Events
- Let \( E_1 \) be the event that the examinee guesses the answer.
- Let \( E_2 \) be the event that the examinee copies the answer.
- Let \( E_3 \) be the event that the examinee knows the answer.

### Step 2: Given Probabilities
From the problem, we have:
- \( P(E_1) = \frac{1}{3} \) (Probability of guessing)
- \( P(E_2) = \frac{1}{6} \) (Probability of copying)
- The probability of knowing the answer can be found as:
\[ P(E_3) = 1 - P(E_1) - P(E_2) = 1 - \frac{1}{3} - \frac{1}{6} = \frac{1}{2} \]

### Step 3: Probability of Correct Answer Given Each Event
- If the examinee guesses, the probability of getting the correct answer is:
\[ P(C|E_1) = \frac{1}{4} \]
- If the examinee copies the answer, the probability of getting the correct answer is:
\[ P(C|E_2) = \frac{1}{8} \]
- If the examinee knows the answer, the probability of getting the correct answer is:
\[ P(C|E_3) = 1 \]

### Step 4: Total Probability of Correct Answer
Using the law of total probability, we can calculate \( P(C) \):
\[ P(C) = P(C|E_1) P(E_1) + P(C|E_2) P(E_2) + P(C|E_3) P(E_3) \]
Substituting the values:
\[ P(C) = \left(\frac{1}{4} \cdot \frac{1}{3}\right) + \left(\frac{1}{8} \cdot \frac{1}{6}\right) + \left(1 \cdot \frac{1}{2}\right) \]
Calculating each term:
\[ P(C) = \frac{1}{12} + \frac{1}{48} + \frac{1}{2} \]
To combine these fractions, we need a common denominator, which is 48:
\[ P(C) = \frac{4}{48} + \frac{1}{48} + \frac{24}{48} = \frac{29}{48} \]

### Step 5: Applying Bayes' Theorem
We want to find the probability that the examinee knew the answer given that he answered correctly, \( P(E_3|C) \):
\[ P(E_3|C) = \frac{P(C|E_3) P(E_3)}{P(C)} \]
Substituting the values:
\[ P(E_3|C) = \frac{1 \cdot \frac{1}{2}}{\frac{29}{48}} = \frac{\frac{1}{2}}{\frac{29}{48}} = \frac{24}{29} \]

### Final Answer
Thus, the probability that the examinee knew the answer given that he answered correctly is:
\[ \boxed{\frac{24}{29}} \]