Probability
Bayes' Theorem
Grade 12
Question:
<p><strong>For Problems 7–9:</strong> A JEE aspirant estimates that she will be successful with an 80% chance if she studies 10 hours per day, with a 60% chance if she studies 7 hours per day, and with a 40% chance if she studies 4 hours per day. She further believes that she will study 10 hours, 7 hours, and 4 hours per day with probabilities 0.1, 0.2, and 0.7, respectively.</p><p>Given that she does not achieve success, the chance she studied for 4 hours is</p>
<p>(1) 18/26</p>
<p>(2) 19/26</p>
<p>(3) 20/26</p>
<p>(4) 21/26</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem with the law of total probability: Given failure (not success), find P(studied 4 hrs | failed) = P(failed | 4 hrs) × P(4 hrs) / P(failed)
<p><strong>Step 1:</strong> Identify given probabilities</p><p>P(success | 10 hrs) = 0.8 → P(failure | 10 hrs) = 0.2</p><p>P(success | 7 hrs) = 0.6 → P(failure | 7 hrs) = 0.4</p><p>P(success | 4 hrs) = 0.4 → P(failure | 4 hrs) = 0.6</p><p>P(10 hrs) = 0.1, P(7 hrs) = 0.2, P(4 hrs) = 0.7</p><p><strong>Step 2:</strong> Calculate total probability of failure using law of total probability</p><p>P(failure) = P(failure | 10 hrs)×P(10 hrs) + P(failure | 7 hrs)×P(7 hrs) + P(failure | 4 hrs)×P(4 hrs)</p><p>P(failure) = 0.2(0.1) + 0.4(0.2) + 0.6(0.7)</p><p>P(failure) = 0.02 + 0.08 + 0.42 = 0.52</p><p><strong>Step 3:</strong> Apply Bayes' theorem</p><p>P(4 hrs | failure) = [P(failure | 4 hrs) × P(4 hrs)] / P(failure)</p><p>P(4 hrs | failure) = (0.6 × 0.7) / 0.52 = 0.42 / 0.52</p><p>P(4 hrs | failure) = 42/52 = 21/26</p><p>∴ Answer: D</p>
Correct Answer: D