Probability
Total Probability 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>The chance she will be successful is</p>
<p>(1) 0.28</p>
<p>(2) 0.38</p>
<p>(3) 0.48</p>
<p>(4) 0.58</p>
Step-by-Step Solution
Key Concept: Use the law of total probability by conditioning on the number of hours studied: P(Success) = Σ P(Success|Hours) × P(Hours). Each study duration is a separate case that must be weighted by its probability.
<p><strong>Step 1:</strong> Identify the conditional probabilities and prior probabilities.</p><p>P(Success | 10 hrs) = 0.80, P(10 hrs) = 0.1<br>P(Success | 7 hrs) = 0.60, P(7 hrs) = 0.2<br>P(Success | 4 hrs) = 0.40, P(4 hrs) = 0.7</p><p><strong>Step 2:</strong> Apply the law of total probability.</p><p>P(Success) = P(Success | 10 hrs) × P(10 hrs) + P(Success | 7 hrs) × P(7 hrs) + P(Success | 4 hrs) × P(4 hrs)</p><p><strong>Step 3:</strong> Substitute the values.</p><p>P(Success) = (0.80)(0.1) + (0.60)(0.2) + (0.40)(0.7)<br>= 0.08 + 0.12 + 0.28<br>= 0.48</p><p>∴ The chance she will be successful is <strong>0.48 or 48%</strong></p>
Correct Answer: C