Question

In an entrance test, there are multiple choice questions. There are four possible answers to each question of which only one is correct. The probability that a student knows the answer to a question is 90%. If he gets the correct answer to a question, then the probability that he was guessing is

• A. \( \frac{36}{37} \)
• B. \( \frac{1}{9} \)
• C. \( \frac{1}{37} \)
• D. \( \frac{37}{40} \)

Hint:

Use Bayes' theorem when finding reverse probability: \(P(A|B)=\frac{P(B|A)P(A)}{P(B)}\).

Answer 1

The Correct Option is C

Step 1: Define events.
Let:
\[ K = \text{student knows the answer}, \quad G = \text{student guesses}. \]
Given:
\[ P(K)=0.9,\quad P(G)=0.1. \]

Step 2: Probability of correct answer.
If student knows the answer:
\[ P(C|K)=1. \]
If student guesses (4 options):
\[ P(C|G)=\frac{1}{4}. \]

Step 3: Find total probability of correct answer.
\[ P(C)=P(C|K)P(K)+P(C|G)P(G). \]
\[ =1\times0.9+\frac{1}{4}\times0.1. \]
\[ =0.9+0.025=0.925. \]

Step 4: Use Bayes' theorem.
\[ P(G|C)=\frac{P(C|G)P(G)}{P(C)}. \]

Step 5: Substitute values.
\[ P(G|C)=\frac{\frac{1}{4}\times0.1}{0.925}. \]
\[ =\frac{0.025}{0.925}. \]

Step 6: Convert into fraction.
\[ 0.025=\frac{1}{40},\quad 0.925=\frac{37}{40}. \]
\[ P(G|C)=\frac{\frac{1}{40}}{\frac{37}{40}}=\frac{1}{37}. \]

Step 7: Final conclusion.
\[ \boxed{\frac{1}{37}} \]