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

Options

A. 2
B. 3
C. 4
D. `sqrt(2)`

Answer

To solve the problem step by step, we will follow the given information about the binomial distribution and apply the necessary formulas.

### Step-by-Step Solution:

1. **Identify Given Information**:
- Number of trials (n) = 9
- Mean (μ) = 6

2. **Recall the Mean of a Binomial Distribution**:
- The mean (μ) of a binomial distribution is given by the formula:
\[ \mu = n \cdot p \]
where \( p \) is the probability of success.

3. **Substitute the Known Values into the Mean Formula**:
- We can rearrange the formula to find \( p \):
\[ p = \frac{\mu}{n} = \frac{6}{9} = \frac{2}{3} \]

4. **Find the Probability of Failure (q)**:
- Since \( q = 1 - p \):
\[ q = 1 - \frac{2}{3} = \frac{1}{3} \]

5. **Recall the Variance of a Binomial Distribution**:
- The variance (σ²) of a binomial distribution is given by the formula:
\[ \sigma^2 = n \cdot p \cdot q \]

6. **Substitute the Values into the Variance Formula**:
- Now we can substitute \( n \), \( p \), and \( q \) into the variance formula:
\[ \sigma^2 = 9 \cdot \frac{2}{3} \cdot \frac{1}{3} \]

7. **Calculate the Variance**:
- Performing the multiplication:
\[ \sigma^2 = 9 \cdot \frac{2}{3} \cdot \frac{1}{3} = 9 \cdot \frac{2}{9} = 2 \]

8. **Conclusion**:
- The variance of the binomial distribution is:
\[ \sigma^2 = 2 \]

### Final Answer:
The variance of the binomial distribution is **2**.

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