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.

Let:
- A = event that the student knows the answer
- B = event that the student guesses the answer
- C = event that the student answers correctly

Given:
- \( P(A) = \frac{3}{4} \) (probability that the student knows the answer)
- \( P(B) = \frac{1}{4} \) (probability that the student guesses)
- \( P(C|A) = 1 \) (if the student knows the answer, he answers correctly)
- \( P(C|B) = \frac{1}{4} \) (if the student guesses, the probability of answering correctly is \( \frac{1}{4} \))

We want to find \( P(A|C) \), the probability that the student knows the answer given that he answered correctly.

Using Bayes' theorem:
\[ P(A|C) = \frac{P(C|A) \cdot P(A)}{P(C)} \]

First, we need to calculate \( P(C) \), the total probability of answering correctly, which can be found using the law of total probability:
\[ P(C) = P(C|A) \cdot P(A) + P(C|B) \cdot P(B) \]

Substituting the 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, we need a common denominator. The least common multiple of 4 and 16 is 16:
\[ P(C) = \frac{3 \cdot 4}{16} + \frac{1}{16} = \frac{12}{16} + \frac{1}{16} = \frac{13}{16} \]

Now we can substitute \( P(C) \) back into Bayes' theorem:
\[ P(A|C) = \frac{P(C|A) \cdot P(A)}{P(C)} = \frac{(1) \cdot \left(\frac{3}{4}\right)}{\frac{13}{16}} \]
\[ P(A|C) = \frac{\frac{3}{4}}{\frac{13}{16}} = \frac{3}{4} \cdot \frac{16}{13} = \frac{48}{52} = \frac{12}{13} \]

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