Answer: 1.3

## Explanation For a binomial random variable X with parameters n (number of trials) and p (probability of success), we have: - **Expected value**: E(X) = np - **Variance**: Var(X) = np(1-p) ### Step 1: Use the given probabilities We are given: - P(X=0) = 0.20 - P(X=1) = 0.35 - E(X) = 1.5 For a binomial distribution: - P(X=0) = (1-p)^n = 0.20 - P(X=1) = n × p × (1-p)^(n-1) = 0.35 ### Step 2: Use the ratio of probabilities Divide P(X=1) by P(X=0): \[ \frac{P(X=1)}{P(X=0)} = \frac{n \times p \times (1-p)^{n-1}}{(1-p)^n} = \frac{n \times p}{1-p} = \frac{0.35}{0.20} = 1.75 \] So: \[ \frac{n \times p}{1-p} = 1.75 \] ### Step 3: Use the expected value We know E(X) = np = 1.5 Substitute np = 1.5 into the ratio: \[ \frac{1.5}{1-p} = 1.75 \] \[ 1.5 = 1.75(1-p) \] \[ 1.5 = 1.75 - 1.75p \] \[ 1.75p = 1.75 - 1.5 \] \[ 1.75p = 0.25 \] \[ p = \frac{0.25}{1.75} = \frac{1}{7} \approx 0.142857 \] ### Step 4: Find n Since np = 1.5 and p = 1/7: \[ n \times \frac{1}{7} = 1.5 \] \[ n = 1.5 \times 7 = 10.5 \] Since n must be an integer, this suggests the distribution may not be exactly binomial, but we can proceed with the variance calculation. ### Step 5: Calculate variance Var(X) = np(1-p) = 1.5 × (1 - 1/7) = 1.5 × (6/7) = 9/7 ≈ 1.2857 This rounds to **1.3**, which matches option D. **Answer: D (1.3)**

Author: Tanishq Prabhu