If the mean of a binomial distribution with 9 trials is 6, then its variance is

Options

A. 2
B. 3
C. 4
D. `sqrt2`

Answer

To solve the problem step by step, we will use the properties of a binomial distribution.

**Step 1: Identify the given values.**
- The number of trials (n) = 9
- The mean (μ) = 6

**Step 2: Use the formula for the mean of a binomial distribution.**
The mean of a binomial distribution is given by the formula:
\[ \text{Mean} (μ) = n \cdot p \]
where \( p \) is the probability of success.

**Step 3: Substitute the known values into the mean formula.**
We know that:
\[ n = 9 \quad \text{and} \quad μ = 6 \]
Substituting these values into the mean formula:
\[ 6 = 9 \cdot p \]

**Step 4: Solve for \( p \).**
To find \( p \), we can rearrange the equation:
\[ p = \frac{6}{9} = \frac{2}{3} \]

**Step 5: Use the formula for the variance of a binomial distribution.**
The variance (σ²) of a binomial distribution is given by the formula:
\[ \text{Variance} (σ^2) = n \cdot p \cdot (1 - p) \]

**Step 6: Substitute the values of \( n \) and \( p \) into the variance formula.**
Now we substitute \( n = 9 \) and \( p = \frac{2}{3} \):
\[ σ^2 = 9 \cdot \frac{2}{3} \cdot \left(1 - \frac{2}{3}\right) \]

**Step 7: Calculate \( 1 - p \).**
\[ 1 - p = 1 - \frac{2}{3} = \frac{1}{3} \]

**Step 8: Substitute \( 1 - p \) back into the variance formula.**
\[ σ^2 = 9 \cdot \frac{2}{3} \cdot \frac{1}{3} \]

**Step 9: Simplify the expression.**
Calculating the right side:
\[ σ^2 = 9 \cdot \frac{2}{3} \cdot \frac{1}{3} = 9 \cdot \frac{2}{9} = 2 \]

**Final Answer:**
The variance of the binomial distribution is \( 2 \).

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