Question

Anubhab faces a multiple-choice question (MCQ) with four alternative choices, where only one is the right choice. There is no negative mark for making a wrong choice. Also, assume that the probability that he knows the answer is 0.50. The probability of making the correct choice is 0.25, if he does not know the answer. He got full marks in this question.
The probability that he knew the answer is ......... (rounded off to one decimal place).

Hint:

Bayes' theorem is a powerful tool for calculating conditional probabilities based on known information and prior probabilities.

Answer 1

Let \( P(K) \) be the probability that Anubhab knew the answer and \( P(\neg K) \) be the probability that he did not know the answer. We are given:
- \( P(K) = 0.50 \) (probability that Anubhab knew the answer),
- \( P(\neg K) = 1 - P(K) = 0.50 \) (probability that Anubhab did not know the answer),
- \( P(\text{correct} | K) = 1 \) (if he knows the answer, he answers correctly),
- \( P(\text{correct} | \neg K) = 0.25 \) (if he does not know the answer, he guesses, with 0.25 probability of answering correctly).
We are asked to find the probability that he knew the answer given that he got the question correct. This is a conditional probability problem, which we can solve using Bayes’ theorem: \[ P(K | \text{correct}) = \frac{P(\text{correct} | K) \cdot P(K)}{P(\text{correct})} \] Step 1: Find \( P(\text{correct}) \), the total probability of answering correctly: \[ P(\text{correct}) = P(\text{correct} | K) \cdot P(K) + P(\text{correct} | \neg K) \cdot P(\neg K) \] \[ P(\text{correct}) = (1 \cdot 0.50) + (0.25 \cdot 0.50) = 0.50 + 0.125 = 0.625 \] Step 2: Apply Bayes’ theorem: \[ P(K | \text{correct}) = \frac{1 \cdot 0.50}{0.625} = \frac{0.50}{0.625} = 0.80 \] Thus, the probability that Anubhab knew the answer is \( 0.80 \). Final Answer: \[ \boxed{0.8} \]