If the Mean and Variance of a Binomial Distribution are 12 and 8 respectively, find the number of trials.

Answer

To solve the problem, we need to find the number of trials \( n \) in a binomial distribution given that the mean is 12 and the variance is 8.

### Step-by-Step Solution:

1. **Understand the Mean and Variance of a Binomial Distribution**:
- The mean \( \mu \) of a binomial distribution is given by the formula:
\[ \mu = n \cdot p \]
- The variance \( \sigma^2 \) of a binomial distribution is given by the formula:
\[ \sigma^2 = n \cdot p \cdot q \]
where \( q = 1 - p \).

2. **Set Up the Equations**:
- From the problem, we know:
\[ n \cdot p = 12 \quad \text{(1)} \]
\[ n \cdot p \cdot q = 8 \quad \text{(2)} \]

3. **Express \( q \) in Terms of \( p \)**:
- Since \( q = 1 - p \), we can substitute \( q \) into equation (2):
\[ n \cdot p \cdot (1 - p) = 8 \]

4. **Substitute \( n \cdot p \) from Equation (1)**:
- From equation (1), we know \( n \cdot p = 12 \). Substitute this into the modified equation (2):
\[ 12 \cdot (1 - p) = 8 \]

5. **Solve for \( p \)**:
- Rearranging gives:
\[ 12 - 12p = 8 \]
\[ 12p = 12 - 8 \]
\[ 12p = 4 \]
\[ p = \frac{4}{12} = \frac{1}{3} \]

6. **Find \( q \)**:
- Now that we have \( p \), we can find \( q \):
\[ q = 1 - p = 1 - \frac{1}{3} = \frac{2}{3} \]

7. **Substitute \( p \) Back to Find \( n \)**:
- Now substitute \( p \) back into equation (1):
\[ n \cdot \frac{1}{3} = 12 \]
\[ n = 12 \cdot 3 = 36 \]

### Final Answer:
The number of trials \( n \) is **36**.

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