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 the answer is correct with a probability of $\frac{1}{3}$. What is the probability that the student knows the answer, given that he has answered it correctly?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → $\frac{9}{11}$ ##
Let $E_1 = \text{Student knows the answer}$
$E_2 = \text{Student guesses the answer}$
$A = \text{Student has answered the question correctly}$
$P(E_1) = \frac{3}{5}, \quad P(E_2) = \frac{2}{5}$
$P(A|E_1) = \text{Probability student answered correctly given he knows the answer} = 1$
$P(A|E_2) = \text{Probability student answered correctly given he guesses} = \frac{1}{3}$
$P\left(\frac{E_1}{A}\right) = \frac{P(E_1) \cdot P\left(\frac{A}{E_1}\right)}{P(E_1) \cdot P\left(\frac{A}{E_1}\right) + P(E_2) \cdot P\left(\frac{A}{E_2}\right)}$
$= \frac{\frac{3}{5} \cdot 1}{\frac{3}{5} \cdot 1 + \frac{2}{5} \cdot \frac{1}{3}} = \frac{\frac{3}{5}}{\frac{3}{5} + \frac{2}{15}}$
$= \frac{\frac{3}{5}}{\frac{9 + 2}{15}} = \frac{3}{5} \cdot \frac{15}{11} = \frac{9}{11}$
Required probability = $\frac{9}{11}$