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:

We start by noting the composition of the box. There are 15 green balls and 10 yellow balls, making a total of

$$15+10 = 25$$

balls in the box. In every single draw (because the ball is replaced), the probability of choosing a green ball remains constant. That probability is simply the ratio of green balls to total balls, namely

$$p = \frac{\text{number of green balls}}{\text{total balls}} = \frac{15}{25} = \frac{3}{5}.$$

Correspondingly, the probability of drawing a yellow ball is

$$q = 1-p = 1-\frac{3}{5} = \frac{2}{5}.$$

Let us introduce random variables to model the situation clearly. Let

$$X_i = \begin{cases}

1, &\text{if the $$i$$-th draw yields a green ball},\\[4pt]

0, &\text{if the $$i$$-th draw yields a yellow ball},

\end{cases}\qquad i = 1,2,\dots ,10.$$

Because each draw is made with replacement, the outcomes of different draws are independent and the probability of success (getting green) stays the same for every draw. Hence each $$X_i$$ is a Bernoulli random variable with

$$P(X_i = 1) = p = \frac{3}{5}, \qquad P(X_i = 0) = q = \frac{2}{5}.$$

The total number of green balls drawn in 10 attempts is then

$$X = X_1 + X_2 + \dots + X_{10}.$$

Now, the sum of independent Bernoulli($$p$$) variables is a Binomial($$n,p$$) random variable. Therefore

$$X \sim \text{Binomial}(n = 10,\; p = \frac{3}{5}).$$

For a Binomial($$n,p$$) distribution, the variance formula is first recalled:

$$\text{Variance of Binomial}(n,p):\quad \operatorname{Var}(X) = n\,p\,q,$$

where $$q = 1-p$$. Substituting the known values $$n = 10,\; p = \frac{3}{5},\; q = \frac{2}{5},$$ we get

$$\operatorname{Var}(X) = 10 \times \frac{3}{5} \times \frac{2}{5}.$$

Multiplying step by step, we have

$$10 \times \frac{3}{5} = \frac{30}{5} = 6,$$

$$6 \times \frac{2}{5} = \frac{12}{5}.$$

Thus, the variance of the number of green balls drawn is

$$\operatorname{Var}(X) = \frac{12}{5}.$$

Among the given options, this value corresponds to Option A.

Hence, the correct answer is Option A.