Question

If a random variable $X$ follows the Binomial distribution $\text{B}(33, \text{p})$ such that $3\text{P}(\text{X} = 0) = \text{P}(\text{X} = 1)$, then the variance of X is

• A. $\frac{11}{144}$
• B. $\frac{35}{48}$
• C. $\frac{121}{48}$
• D. $\frac{33}{144}$

Hint:

Use ratio of probabilities to find \(p\) quickly.

Answer 1

The Correct Option is C

Concept:
For binomial distribution: \[ P(X=0) = (1-p)^{33}, \quad P(X=1) = 33p(1-p)^{32} \] Step 1: Given condition. \[ 3(1-p)^{33} = 33p(1-p)^{32} \]
Step 2: Simplify. \[ 3(1-p) = 33p \] \[ 3 = 36p \Rightarrow p = \frac{1}{12} \]
Step 3: Find variance. \[ \text{Var}(X) = npq = 33 \cdot \frac{1}{12} \cdot \frac{11}{12} \] \[ = \frac{33 \times 11}{144} = \frac{363}{144} = \frac{121}{48} \]
Step 4: Conclusion. \[ \text{Variance} = \frac{121}{48} \]