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 is successful, the chance that she studied for 4 hours is</p>
<p>(1) 6/12</p>
<p>(2) 7/12</p>
<p>(3) 8/12</p>
<p>(4) 9/12</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem to find P(4 hours | success) by calculating the posterior probability. The denominator requires the law of total probability to find P(success) across all study hours.
<p><strong>Step 1:</strong> Identify given information.</p><p>P(10 hrs) = 0.1, P(7 hrs) = 0.2, P(4 hrs) = 0.7</p><p>P(success | 10 hrs) = 0.8, P(success | 7 hrs) = 0.6, P(success | 4 hrs) = 0.4</p><p><strong>Step 2:</strong> Find total probability of success using law of total probability.</p><p>P(success) = P(success | 10) · P(10) + P(success | 7) · P(7) + P(success | 4) · P(4)</p><p>P(success) = 0.8(0.1) + 0.6(0.2) + 0.4(0.7)</p><p>P(success) = 0.08 + 0.12 + 0.28 = 0.48</p><p><strong>Step 3:</strong> Apply Bayes' theorem to find P(4 hrs | success).</p><p>P(4 hrs | success) = [P(success | 4 hrs) · P(4 hrs)] / P(success)</p><p>P(4 hrs | success) = [0.4 × 0.7] / 0.48</p><p>P(4 hrs | success) = 0.28 / 0.48 = 28/48 = 7/12</p><p>∴ Answer: B</p>
Correct Answer: B