Answer: 0.023

## Explanation For a binomial random variable X with parameters n (number of trials) and p (probability of success): 1. **Expected value**: E(X) = np = 2 2. **Variance**: Var(X) = np(1-p) = 1.5 From the variance formula: Var(X) = np(1-p) = 2(1-p) = 1.5 Solving for p: 2(1-p) = 1.5 1-p = 0.75 p = 0.25 Now, using E(X) = np = 2: n × 0.25 = 2 n = 8 So we have n = 8, p = 0.25 **Calculating P(X = 5)**: P(X = 5) = C(8,5) × (0.25)^5 × (0.75)^3 Where C(8,5) = 8!/(5!×3!) = 56 P(X = 5) = 56 × (0.25)^5 × (0.75)^3 = 56 × 0.0009765625 × 0.421875 = 56 × 0.0004119873046875 = 0.0230712890625 ≈ 0.023 Therefore, the correct answer is **B. 0.023**.

Author: Nikitesh Somanthe