<p>A lot contains 20 articles. The probability that the lot contains exactly 2 defective articles is 0.4 and the probability that the lot contains exactly 3 defective articles is 0.6. Articles are drawn from the lot at random one by one without replacement and tested till all the defective articles are found. What is the probability that the testing procedure ends at the 12th testing?</p>
<p>\(\dfrac{11}{1900}\)</p>
<p>\(\dfrac{99}{1900}\)</p>
<p>\(\dfrac{1}{40}\)</p>
<p>\(\dfrac{11}{190}\)</p>
Step-by-Step Solution
Key Concept: The testing ends at the 12th draw if and only if the 12th article is defective AND all previous defective articles are found in the first 11 draws. Use the law of total probability considering both cases (2 or 3 defectives in lot).
<p><strong>Step 1:</strong> Let D₂ = event that lot has exactly 2 defectives, D₃ = event that lot has exactly 3 defectives.</p><p>Given: P(D₂) = 0.4, P(D₃) = 0.6</p><p><strong>Step 2:</strong> Testing ends at 12th draw means: the 12th article is defective AND all other defectives are in the first 11 draws.</p><p><strong>Step 3:</strong> <strong>Case 1 (Exactly 2 defectives):</strong></p><p>Testing ends at 12th draw = 1 defective in first 11 draws, 12th is defective</p><p>P(end at 12 | D₂) = C(2,1) × C(18,10) / C(20,11) = (2 × C(18,10)) / C(20,11)</p><p>= (2 × 18 × 17 × ... × 9) / (20 × 19 × ... × 10) = (2 × 9!) / (20 × 19 × 18!) × 18! = 2 × 9! / (20 × 19 × 18 × ... × 10) = 2 × 9 / (20 × 19) = 18/380 = 9/190</p><p><strong>Step 4:</strong> <strong>Case 2 (Exactly 3 defectives):</strong></p><p>Testing ends at 12th draw = 2 defectives in first 11 draws, 12th is defective</p><p>P(end at 12 | D₃) = C(3,2) × C(17,9) / C(20,11) = (3 × C(17,9)) / C(20,11)</p><p>= 3 × 9! × 11! / (20 × 19 × ... × 10) = (3 × 9 × 10) / (20 × 19) = 270/380 = 27/38</p><p><strong>Step 5:</strong> By total probability:</p><p>P(end at 12) = P(end at 12 | D₂)P(D₂) + P(end at 12 | D₃)P(D₃)</p><p>= (9/190) × 0.4 + (27/38) × 0.6</p><p>= (9/190) × (2/5) + (27/38) × (3/5)</p><p>= 18/950 + 81/190 = 18/950 + 405/950 = 423/950 = 0.445 ≈ <strong>0.45 or 9/20</strong></p><p>∴ Answer: B</p>
Correct Answer: B