Official solution
To solve this problem, we need to calculate the probability of two independent events occurring in succession: the first marble drawn is red, and the second marble drawn is white. Since the drawing is with replacement, the number of marbles of each color remains the same for both draws.
The total number of marbles in the box is the sum of red, white, blue, and orange marbles:
\[
\text{Total marbles} = 10 (\text{red}) + 30 (\text{white}) + 20 (\text{blue}) + 15 (\text{orange}) = 75.
\]
The probability of drawing a red marble in the first draw is the number of red marbles divided by the total number of marbles:
\[
P(\text{First is red}) = \frac{10}{75}.
\]
Since the marble is replaced, the probability of drawing a white marble in the second draw remains as the number of white marbles divided by the total number of marbles:
\[
P(\text{Second is white}) = \frac{30}{75}.
\]
The probability of both independent events occurring in succession (drawing a red marble first and then a white marble) is the product of their individual probabilities:
\[
P(\text{First is red and second is white}) = P(\text{First is red}) \times P(\text{Second is white}) = \frac{10}{75} \times \frac{30}{75}.
\]
Now, let's calculate this probability:
\[
P(\text{First is red and second is white}) = \frac{10 \times 30}{75 \times 75} = \frac{300}{5625}= \frac{4}{75}.
\]
Therefore, the correct answer is
Option D
\[\frac{4}{75}.\]