A box contains `15` green and `10` yellow balls. If `10` balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is : (a) `12/5` (b) `6` (c) `4` (d) `6/25`

Options

A. `(6)/(25)`
B. `(12)/(5)`
C. 6
D. 4

Answer

To solve the problem, we need to find the variance of the number of green balls drawn when 10 balls are drawn from a box containing 15 green and 10 yellow balls, with replacement.

### Step-by-Step Solution:

1. **Determine the Total Number of Balls:**
- Total green balls = 15
- Total yellow balls = 10
- Total balls = 15 + 10 = 25

2. **Calculate the Probability of Drawing a Green Ball:**
- Probability of drawing a green ball (p) = Number of green balls / Total number of balls
- \( p = \frac{15}{25} = \frac{3}{5} \)

3. **Calculate the Probability of Drawing a Yellow Ball:**
- Probability of drawing a yellow ball (q) = Number of yellow balls / Total number of balls
- \( q = \frac{10}{25} = \frac{2}{5} \)

4. **Identify the Number of Trials:**
- Number of balls drawn (n) = 10

5. **Use the Variance Formula:**
- The variance of a binomial distribution is given by the formula:
\[ \text{Variance} = n \cdot p \cdot q \]
- Substituting the values we have:
\[ \text{Variance} = 10 \cdot \frac{3}{5} \cdot \frac{2}{5} \]

6. **Calculate the Variance:**
- First, calculate \( p \cdot q \):
\[ p \cdot q = \frac{3}{5} \cdot \frac{2}{5} = \frac{6}{25} \]
- Now, multiply by n:
\[ \text{Variance} = 10 \cdot \frac{6}{25} = \frac{60}{25} = \frac{12}{5} \]

7. **Final Answer:**
- The variance of the number of green balls drawn is \( \frac{12}{5} \).

### Conclusion:
The correct option is (a) \( \frac{12}{5} \).