In answering a question on a multiple choice test a student either knows the answer or guesses. Let the probability that he knows the answer is `3/4` and probability that he guesses is `1/4`. Assuming that a student who guesses the answer and given correct answer is `1/4`. What is the probability that student knows the answer given that he answered it correctly?

Explanation

To solve the problem, we will use Bayes' theorem. We need to find the probability that the student knows the answer given that he answered it correctly, denoted as \( P(A|C) \).

### Step-by-step Solution:

1. **Define Events:**
- Let \( A \): the event that the student knows the answer.
- Let \( B \): the event that the student guesses the answer.
- Let \( C \): the event that the student answers correctly.

2. **Given Probabilities:**
- \( P(A) = \frac{3}{4} \) (the probability that the student knows the answer).
- \( P(B) = \frac{1}{4} \) (the probability that the student guesses the answer).
- \( P(C|B) = \frac{1}{4} \) (the probability of answering correctly given that the student guesses).
- \( P(C|A) = 1 \) (the probability of answering correctly given that the student knows the answer).

3. **Apply Bayes' Theorem:**
Bayes' theorem states:
\[ P(A|C) = \frac{P(C|A) \cdot P(A)}{P(C)} \]
We need to find \( P(C) \) first.

4. **Calculate \( P(C) \):**
Using the law of total probability:
\[ P(C) = P(C|A) \cdot P(A) + P(C|B) \cdot P(B) \]
Substitute the known values:
\[ P(C) = (1) \cdot \left(\frac{3}{4}\right) + \left(\frac{1}{4}\right) \cdot \left(\frac{1}{4}\right) \]
\[ P(C) = \frac{3}{4} + \frac{1}{16} \]
To add these fractions, find a common denominator (16):
\[ P(C) = \frac{3 \cdot 4}{16} + \frac{1}{16} = \frac{12 + 1}{16} = \frac{13}{16} \]

5. **Substitute Back into Bayes' Theorem:**
Now substitute \( P(C) \) back into the equation for \( P(A|C) \):
\[ P(A|C) = \frac{P(C|A) \cdot P(A)}{P(C)} = \frac{1 \cdot \frac{3}{4}}{\frac{13}{16}} \]
Simplifying this:
\[ P(A|C) = \frac{\frac{3}{4}}{\frac{13}{16}} = \frac{3}{4} \cdot \frac{16}{13} = \frac{3 \cdot 16}{4 \cdot 13} = \frac{48}{52} = \frac{12}{13} \]

### Final Answer:
Thus, the probability that the student knows the answer given that he answered it correctly is:
\[ \boxed{\frac{12}{13}} \]