Probability
Bayes' Theorem
Grade 12
Question:
<p>In an entrance test, there are multiple choice questions. There are four possible answers to each question, of which 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 find the probability that he was guessing.</p>
<p>\(\dfrac{1}{37}\)</p>
<p>\(\dfrac{2}{37}\)</p>
<p>\(\dfrac{1}{27}\)</p>
<p>\(\dfrac{1}{7}\)</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem to find P(guessing | correct answer). The student can get correct either by knowing (90%) or by lucky guessing (10% × 1/4).
<p><strong>Step 1:</strong> Define events:</p><p>K = student knows the answer, G = student guesses</p><p>P(K) = 0.9, P(G) = 0.1</p><p><strong>Step 2:</strong> Find probability of correct answer by each method:</p><p>P(Correct|K) = 1 (if knows, always correct)</p><p>P(Correct|G) = 1/4 (if guesses, one correct out of 4 choices)</p><p><strong>Step 3:</strong> Find total probability of getting correct answer:</p><p>P(Correct) = P(Correct|K)·P(K) + P(Correct|G)·P(G)</p><p>P(Correct) = 1 × 0.9 + (1/4) × 0.1 = 0.9 + 0.025 = 0.925</p><p><strong>Step 4:</strong> Apply Bayes' theorem to find P(G|Correct):</p><p>P(G|Correct) = P(Correct|G)·P(G) / P(Correct)</p><p>P(G|Correct) = (1/4 × 0.1) / 0.925 = 0.025 / 0.925 = 25/925 = 1/37</p><p>∴ Answer: <strong>1/37</strong></p>
Correct Answer: A