Problem Statement
In answering a question on a multiple-choice test, a student either knows the answer or guesses. Let $\frac{3}{5}$ be the probability that he knows the answer and $\frac{2}{5}$ be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability $\frac{1}{4}$.
What is the probability that the student knew the answer given that he answered it correctly?
Verified Solution & Marking Scheme
Define events and state prior probabilities
Let $$E_1$$: Student knows the answer, $$E_2$$: Student guesses the answer, $$A$$: Student answers the question correctly.
$$P(E_1) = \frac{3}{5}, \quad P(E_2) = \frac{2}{5}$$
Determine conditional probabilities
If the student knows the answer, he is guaranteed to answer correctly: $$P(A \mid E_1) = 1$$.
If he guesses, the probability of correct answer is $$P(A \mid E_2) = \frac{1}{4}$$.
Apply Bayes' Theorem
$$P(E_1 \mid A) = \frac{P(E_1)P(A \mid E_1)}{P(E_1)P(A \mid E_1) + P(E_2)P(A \mid E_2)}$$
$$= \frac{\frac{3}{5} \cdot 1}{\frac{3}{5} \cdot 1 + \frac{2}{5} \cdot \frac{1}{4}} = \frac{\frac{3}{5}}{\frac{3}{5} + \frac{1}{10}} = \frac{\frac{6}{10}}{\frac{7}{10}} = \frac{6}{7}$$