Probability
Binomial Distribution
Grade 12
Question:
<p>A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is</p>
<p>\(\dfrac{6}{25}\)</p>
<p>\(\dfrac{12}{5}\)</p>
<p>6</p>
<p>4</p>
Step-by-Step Solution
Key Concept: For sampling with replacement, the number of green balls follows a Binomial distribution B(n,p). The variance of a binomial distribution is np(1-p), where n is the number of trials and p is the probability of success.
<p><strong>Step 1:</strong> Identify the distribution type. Since balls are drawn with replacement, we have a Binomial distribution with n=10 trials.</p><p><strong>Step 2:</strong> Find probability of drawing a green ball. Total balls = 15 + 10 = 25. Probability p = 15/25 = 3/5.</p><p><strong>Step 3:</strong> Calculate q = 1 - p = 1 - 3/5 = 2/5.</p><p><strong>Step 4:</strong> Apply binomial variance formula: Var(X) = np(1-p) = npq.</p><p>Var(X) = 10 × (3/5) × (2/5) = 10 × 6/25 = 60/25 = 12/5 = 2.4</p><p>∴ Answer: B</p>
Correct Answer: B