Question

If the mean and variance of a binomial distribution are \( 18 \) and \( 12 \) respectively, then the value of \( n \) is __________.

• A. \( 36 \)
• B. \( 54 \)
• C. \( 18 \)
• D. \( 27 \)

Hint:

For a binomial distribution, the formulas for mean and variance are: \[ \mu = np, \quad \sigma^2 = np(1 - p) \] Use these to solve for \( n \) and \( p \).

Answer 1

The Correct Option is B

Step 1: Review of Mean and Variance Formulas
For a binomial distribution with parameters \( n \) and \( p \), the mean and variance are expressed as: \[ \mu = np, \quad \sigma^2 = np(1 - p) \] Step 2: Formulating the Equations
We are provided with the following equations: \[ np = 18, \quad np(1 - p) = 12 \] By dividing the second equation by the first: \[ (1 - p) = \frac{12}{18} = \frac{2}{3} \] which simplifies to: \[ p = \frac{1}{3} \] Step 3: Solving for \( n \)
Substitute \( p = \frac{1}{3} \) into the first equation: \[ n \times \frac{1}{3} = 18 \] Solving for \( n \): \[ n = 18 \times 3 = 54 \]