Probability
Classical Probability
Grade 12
Question:
<p><strong>For Problems 4–6:</strong> In an objective paper, there are two sections of 10 questions each. For 'section 1', each question has 5 options and only one option is correct and 'section 2' has 4 options with multiple answers and marks for a question in this section is awarded only if he ticks all correct answers. Marks for each question in 'section 1' is 1 and in 'section 2' is 3. (There is no negative marking.)</p><p>If a candidate in total attempts 4 questions all by guessing, then the probability of scoring 10 marks is</p>
<p>(1) \(\frac{1}{15}\left(\frac{1}{15}\right)^3\)</p>
<p>(2) \(\frac{4}{5}\left(\frac{1}{15}\right)^3\)</p>
<p>(3) \(\frac{1}{5}\left(\frac{14}{15}\right)^3\)</p>
<p>(4) none of these</p>
Step-by-Step Solution
Key Concept: To score exactly 10 marks by guessing 4 questions, identify all combinations of Section 1 and Section 2 questions that sum to 10 marks, then calculate probability for each valid combination.
<p><strong>Step 1: Identify mark combinations for 10 marks from 4 questions</strong></p><p>Section 1: 1 mark each | Section 2: 3 marks each (requires ALL correct answers)</p><p>Possible cases:</p><ul><li>Case A: 1 question from Section 2 (3 marks) + 7 from Section 1 (7 marks) - IMPOSSIBLE (only 4 questions attempted)</li><li>Case B: 0 questions from Section 2 + all 4 from Section 1 = 4 marks - IMPOSSIBLE</li><li>Case C: 1 question from Section 2 + 3 from Section 1 = 3 + 3 = 6 marks - IMPOSSIBLE</li><li>Case D: 2 questions from Section 2 + 2 from Section 1 = 6 + 2 = 8 marks - IMPOSSIBLE</li><li>Case E: 3 questions from Section 2 + 1 from Section 1 = 9 + 1 = 10 marks ✓</li><li>Case F: 4 questions from Section 2 + 0 from Section 1 = 12 marks - IMPOSSIBLE</li></ul><p><strong>Step 2: Calculate probability for valid case (3 Section 2 + 1 Section 1)</strong></p><p>Ways to choose 3 from Section 2 and 1 from Section 1: C(10,3) × C(10,1) = 120 × 10 = 1200</p><p>Probability of guessing correctly:</p><ul><li>Section 1 (1 question): P = 1/5</li><li>Section 2 (3 questions, all correct): P = (1/2⁴)³ = (1/16)³ for each question</li></ul><p><strong>Step 3: Calculate total probability</strong></p><p>P(10 marks) = C(10,3) × C(10,1) × (1/5) × (1/16)³</p><p>P(10 marks) = 1200 × (1/5) × (1/4096)</p><p>P(10 marks) = 1200/(5 × 4096) = 240/4096 = 15/256</p><p>∴ Answer: D</p>
Correct Answer: D