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>The probability of getting a score less than 40 by answering all the questions by guessing in this paper is</p>
<p>(1) \((1/75)^{10}\)</p>
<p>(2) \(1-(1/75)^{10}\)</p>
<p>(3) \((74/75)^{10}\)</p>
<p>(4) none of these</p>
Step-by-Step Solution
Key Concept: Calculate the probability of scoring <40 by finding the complement: P(score <40) = 1 - P(score ≥40). Section 1 contributes 0-10 marks (1 per question), Section 2 contributes 0-30 marks (3 per question, only if all options correct). You need P(S1 + S2 < 40) where S1+S2 ≥40 requires either high Section 1 scores or perfect Section 2 answers.
<p><strong>Step 1:</strong> Identify scoring structure.</p><p>Section 1: 10 questions, 1 mark each, P(correct by guessing) = 1/5. Maximum = 10 marks.</p><p>Section 2: 10 questions, 3 marks each (only if ALL options ticked correctly), P(correct by guessing) = 1/2⁴ = 1/16. Maximum = 30 marks.</p><p>Total maximum = 40 marks.</p><p><strong>Step 2:</strong> Recognize the key insight.</p><p>P(score < 40) = 1 - P(score = 40)</p><p>To score exactly 40, must get ALL 10 questions in Section 1 correct AND ALL 10 questions in Section 2 correct.</p><p><strong>Step 3:</strong> Calculate P(score = 40).</p><p>P(all Section 1 correct) = (1/5)¹⁰</p><p>P(all Section 2 correct) = (1/16)¹⁰</p><p>P(score = 40) = (1/5)¹⁰ × (1/16)¹⁰ = (1/80)¹⁰</p><p><strong>Step 4:</strong> Find P(score < 40).</p><p>P(score < 40) = 1 - (1/80)¹⁰</p><p>Since (1/80)¹⁰ is negligibly small (≈ 10⁻¹⁹), the answer is approximately <strong>1</strong> or exactly <strong>1 - (1/80)¹⁰</strong>.</p><p>∴ Answer: D</p>
Correct Answer: D