Statistics
Variance / Binomial Distribution
Grade 11
Question:
<p>A box contains 15 green balls and 10 yellow balls (total 25 balls). If 10 balls are drawn at random, find the variance of the number of green balls drawn. (Given \(\sigma^2 = npq\), \(n = 10\), \(p = \dfrac{15}{25} = \dfrac{3}{5}\), \(q = \dfrac{10}{25} = \dfrac{2}{5}\).)</p>
<p>\(\dfrac{6}{5}\)</p>
<p>\(\dfrac{12}{25}\)</p>
<p>\(\dfrac{6}{25}\)</p>
<p>\(\dfrac{12}{5}\)</p>
Step-by-Step Solution
Key Concept: This is sampling without replacement (hypergeometric distribution), NOT a binomial distribution. Students must recognize that the variance formula σ² = npq applies only to binomial cases where sampling is WITH replacement or the population is very large relative to sample size.
<p><strong>Step 1: Identify the distribution type</strong></p><p>Drawing 10 balls from 25 without replacement is <strong>hypergeometric</strong>, not binomial. The draws are NOT independent.</p><p><strong>Step 2: Apply the finite population correction</strong></p><p>For hypergeometric distribution, variance is:</p><p>σ² = npq × (N - n)/(N - 1)</p><p>where:</p><ul><li>n = 10 (balls drawn)</li><li>p = 15/25 = 3/5 (probability of green)</li><li>q = 2/5 (probability of yellow)</li><li>N = 25 (total balls)</li></ul><p><strong>Step 3: Calculate</strong></p><p>npq = 10 × (3/5) × (2/5) = 10 × 6/25 = 2.4</p><p>Correction factor = (N - n)/(N - 1) = (25 - 10)/(25 - 1) = 15/24 = 5/8</p><p><strong>Step 4: Final variance</strong></p><p>σ² = 2.4 × (5/8) = 12/5 × 5/8 = <strong>1.5</strong></p><p>∴ Answer: D (Variance = 1.5)</p>
Correct Answer: D