In an entrance examination, a candidate is to answer a multiple choice question which has four alternative solutions. The candidate either guesses the answer or he knows the answer. The probability that he knows the answer is 0.6. The examiner has found that the candidate has answered the question correctly. Find the probability that the candidate knew the answer, given that he answered it correctly.

Answer

To solve the problem, we will use Bayes' theorem to find the probability that the candidate knew the answer given that he answered correctly.

### Step-by-Step Solution:

1. **Define Events**:
- Let \( A \) be the event that the candidate knows the answer.
- Let \( B \) be the event that the candidate answers the question correctly.

2. **Given Probabilities**:
- The probability that the candidate knows the answer is given as:
\[ P(A) = 0.6 \]
- The probability that the candidate does not know the answer is:
\[ P(A') = 1 - P(A) = 0.4 \]

3. **Conditional Probabilities**:
- If the candidate knows the answer, he will definitely answer correctly:
\[ P(B|A) = 1 \]
- If the candidate does not know the answer, he guesses among 4 options, so the probability of answering correctly is:
\[ P(B|A') = \frac{1}{4} = 0.25 \]

4. **Using Bayes' Theorem**:
- We want to find \( P(A|B) \), the probability that the candidate knew the answer given that he answered correctly. According to Bayes' theorem:
\[ P(A|B) = \frac{P(A) \cdot P(B|A)}{P(A) \cdot P(B|A) + P(A') \cdot P(B|A')} \]

5. **Substituting the Values**:
- Substitute the known values into the formula:
\[ P(A|B) = \frac{0.6 \cdot 1}{0.6 \cdot 1 + 0.4 \cdot 0.25} \]

6. **Calculating the Denominator**:
- Calculate \( P(A) \cdot P(B|A) + P(A') \cdot P(B|A') \):
\[ 0.6 \cdot 1 + 0.4 \cdot 0.25 = 0.6 + 0.1 = 0.7 \]

7. **Final Calculation**:
- Now substitute back into the equation:
\[ P(A|B) = \frac{0.6}{0.7} \]
- Simplifying gives:
\[ P(A|B) = \frac{6}{7} \approx 0.8571 \]

### Conclusion:
The probability that the candidate knew the answer given that he answered correctly is \( \frac{6}{7} \) or approximately 0.8571.