Question

A student answers a multiple choice question with 5 alternatives, of which exactly one is correct. The probability that he knows the correct answer is \( p \), \( 0 < p < 1 \). If he does not know the correct answer, he randomly ticks one answer. Given that he has answered the question correctly, the probability that he did not tick the answer randomly is:

• A. \( \frac{3p}{4p + 3} \)
• B. \( \frac{5p}{3p + 2} \)
• C. \( \frac{5p}{4p + 1} \)
• D. \( \frac{4p}{3p + 1} \)

Hint:

When solving problems using Bayes' Theorem, break down the problem into conditional probabilities and use the law of total probability to find \( P(B) \).

Answer 1

The Correct Option is C

We are given that the student has answered correctly. We want to find the probability that he did not tick the answer randomly, i.e., he knew the correct answer. Let \( A \) be the event that the student knows the correct answer, and \( B \) be the event that the student answers correctly. Step 1: Use Bayes' Theorem Bayes' Theorem gives the probability of \( A \) given \( B \) as: \[ P(A \mid B) = \frac{P(B \mid A) P(A)}{P(B)} \] Step 2: Calculate the necessary probabilities - \( P(A) = p \), the probability that he knows the correct answer. - \( P(B \mid A) = 1 \), the probability of answering correctly given that he knows the answer. - \( P(B \mid A^c) = \frac{1}{5} \), the probability of answering correctly when he does not know the answer, as he randomly guesses. - \( P(A^c) = 1 - p \), the probability that he does not know the answer. Step 3: Calculate \( P(B) \) Using the law of total probability: \[ P(B) = P(B \mid A) P(A) + P(B \mid A^c) P(A^c) = p + \frac{1}{5} (1 - p) \] \[ P(B) = p + \frac{1 - p}{5} = \frac{5p + 1 - p}{5} = \frac{4p + 1}{5} \] Step 4: Apply Bayes' Theorem Now, applying Bayes' Theorem: \[ P(A \mid B) = \frac{1 \times p}{\frac{4p + 1}{5}} = \frac{5p}{4p + 1} \] Thus, the correct answer is \( \frac{5p}{4p + 1} \).