<p>The mean and variance of a random variable having a binomial distribution are 4 and 2, respectively, then \(P(X=1)\) is</p>
Step-by-Step Solution
Key Concept: For binomial distribution, mean = np and variance = np(1-p). Use these two equations to find n and p, then apply the binomial probability formula P(X=k) = C(n,k)p^k(1-p)^(n-k).
<p><strong>Step 1:</strong> Set up equations from given mean and variance.</p><p>For binomial distribution: Mean = np = 4 and Variance = np(1-p) = 2</p><p><strong>Step 2:</strong> Find p by dividing variance by mean.</p><p>np(1-p)/np = 2/4</p><p>1-p = 1/2</p><p>∴ p = 1/2</p><p><strong>Step 3:</strong> Find n using mean equation.</p><p>np = 4</p><p>n(1/2) = 4</p><p>∴ n = 8</p><p><strong>Step 4:</strong> Calculate P(X=1) using binomial probability formula.</p><p>P(X=1) = C(8,1) × (1/2)¹ × (1/2)⁷</p><p>P(X=1) = 8 × (1/2)⁸</p><p>P(X=1) = 8/256 = 1/32</p><p><strong>∴ Answer: P(X=1) = 1/32</strong></p>
Correct Answer: A