We can solve this problem using Bayes' theorem. Let's define the following events :

$A$: The student knows the answer of a randomly chosen question.

$B$: The student guesses the answer of a randomly chosen question.

$C$: The student gives the correct answer to a question.

We are asked to find the probability that the student knows the answer of a randomly chosen question, i.e., $P(A | C)$. According to Bayes' theorem,

$$ P(A | C) = \frac{P(C | A) \cdot P(A)}{P(C)} $$

We need to find the values of $P(C | A)$, $P(A)$, and $P(C)$. Let's proceed step by step :

1. $P(C | A)$ is the probability that the student gives the correct answer given that he knows the answer. Since he knows the answer, he will definitely give the correct answer. Therefore,

$$ P(C | A) = 1 $$

2. $P(A)$ is the probability that the student knows the answer of a randomly chosen question. We don't have this value directly, and we will denote it as $P(A)$. Likewise, the probability that the student guesses the answer is $P(B) = 1 - P(A)$.

3. $P(C)$ is the total probability that the student gives the correct answer. This can be calculated using the Law of Total Probability :

$$ P(C) = P(C | A) \cdot P(A) + P(C | B) \cdot P(B) $$

Where:

$$ P(C | B) = \frac{1}{2} $$

(The probability that the student gives the correct answer given that he guessed it), and

$$ P(B | C) = \frac{1}{6} $$

(The probability that the answer was guessed given the student's answer is correct).

We also have the relation :

$$ P(B | C) = \frac{P(C | B) \cdot P(B)}{P(C)} $$

Substitute the known values and simplify :

$$ \frac{1}{6} = \frac{\frac{1}{2} \cdot P(B)}{P(C)} $$

Solving for $P(C)$ gives :

$$ P(C) = 3 \cdot P(B) $$

From $P(B) = 1 - P(A)$, we get :

$$ P(C) = 3 \cdot (1 - P(A)) $$

We also know from our Law of Total Probability calculation :

$$ P(C) = P(A) + \frac{1}{2} \cdot (1 - P(A)) $$

Equate the two expressions for $P(C)$ :

$$ P(A) + \frac{1}{2} \cdot (1 - P(A)) = 3 \cdot (1 - P(A)) $$

Simplify this equation:

$$ P(A) + \frac{1}{2} - \frac{1}{2} \cdot P(A) = 3 - 3 \cdot P(A) $$

$$ P(A) - \frac{1}{2} \cdot P(A) + \frac{1}{2} = 3 - 3 \cdot P(A) $$

$$ \frac{1}{2} \cdot P(A) + \frac{1}{2} = 3 - 3 \cdot P(A) $$

$$ \frac{7}{2} \cdot P(A) = \frac{5}{2} $$

$$ P(A) = \frac{5}{7} $$

Thus, the probability that the student knows the answer of a randomly chosen question is :

$$ \boxed{\frac{5}{7}} $$