<p>In a box containing 100 bulbs, 10 are defective. What is the probability that out of a sample of 5 bulbs, none is defective?</p>
<p>(a) \(10^{-5}\)</p>
<p>(b) \(\left(\frac{1}{2}\right)^5\)</p>
<p>(c) \(\left(\frac{9}{10}\right)^5\)</p>
<p>(d) \(\frac{9}{10}\)</p>
Step-by-Step Solution
Key Concept: Use hypergeometric distribution or combination principle: the probability equals the ratio of ways to choose 5 non-defective bulbs to total ways of choosing any 5 bulbs from 100.
<p><strong>Step 1:</strong> Identify the setup: 100 bulbs total, 10 defective, 90 non-defective. We select 5 bulbs without replacement.</p><p><strong>Step 2:</strong> Calculate favorable outcomes (all 5 non-defective): C(90,5) = 90!/(5!×85!)</p><p><strong>Step 3:</strong> Calculate total possible outcomes: C(100,5) = 100!/(5!×95!)</p><p><strong>Step 4:</strong> Apply hypergeometric probability:</p><p>P(none defective) = C(90,5)/C(100,5) = [90×89×88×87×86]/[100×99×98×97×96]</p><p><strong>Step 5:</strong> Simplify the fraction: = (90×89×88×87×86)/(100×99×98×97×96) ≈ 0.5839 or approximately 58.39%</p><p><strong>Step 6:</strong> If options are given in exact form, the answer is C(90,5)/C(100,5) or the decimal/percentage equivalent ≈ 0.584</p><p>∴ Answer: C</p>
Correct Answer: C