Probability
Total Probability and Bayes' Theorem
Grade 12
Question:
<p>In a test, an examinee either guesses or copies or knows the answer to a multiple-choice question with four choices, only one answer being correct. The probability that he makes a guess is \(\dfrac{1}{3}\) and the probability that he copies the answer is \(\dfrac{1}{6}\). The probability that his answer is correct, given that he copies it, is \(\dfrac{1}{8}\). Find the probability that he knew the answer to the question, given that he correctly answers.</p>
<p>\(\dfrac{24}{29}\)</p>
<p>\(\dfrac{66}{300}\)</p>
<p>\(\dfrac{11}{50}\)</p>
<p>\(\dfrac{1}{2}\)</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem to find P(knows | correct) by first calculating P(correct) using the law of total probability across three mutually exclusive cases: guessing, copying, and knowing.
<p><strong>Step 1:</strong> Identify given probabilities.</p><p>P(guesses) = 1/3, P(copies) = 1/6, P(knows) = 1 - 1/3 - 1/6 = 1/2</p><p>P(correct | copies) = 1/8, P(correct | guesses) = 1/4 (random choice), P(correct | knows) = 1</p><p><strong>Step 2:</strong> Calculate total probability of correct answer using law of total probability.</p><p>P(correct) = P(correct | guesses)·P(guesses) + P(correct | copies)·P(copies) + P(correct | knows)·P(knows)</p><p>P(correct) = (1/4)·(1/3) + (1/8)·(1/6) + (1)·(1/2)</p><p>P(correct) = 1/12 + 1/48 + 1/2 = 4/48 + 1/48 + 24/48 = 29/48</p><p><strong>Step 3:</strong> Apply Bayes' theorem.</p><p>P(knows | correct) = P(correct | knows)·P(knows) / P(correct)</p><p>P(knows | correct) = (1)·(1/2) / (29/48) = (1/2)·(48/29) = 24/29</p><p>∴ Answer: A</p>
Correct Answer: A