Key Concepts and Formulas

• Probability of an Event: The probability of an event $E$ occurring is defined as the ratio of the number of favorable outcomes for $E$ to the total number of possible outcomes in the sample space. $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$
• Independent Events: Two events, A and B, are considered independent if the occurrence or non-occurrence of one event does not affect the probability of the other event occurring. In this problem, drawing marbles "with replacement" ensures that each successive draw is an independent event, as the original conditions (total marbles, number of each color) are restored.
• Probability of Independent Events Occurring Together: If events A and B are independent, the probability that both A and B occur (denoted as $P(A \text{ and } B)$ or $P(A \cap B)$) is the product of their individual probabilities: $P(A \text{ and } B) = P(A) \times P(B)$

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Step-by-Step Solution

Step 1: Determine the Total Number of Marbles First, we need to establish the size of our sample space, which is the total number of marbles in the box. This sum represents all possible outcomes for any single draw.

• Number of red marbles = 10
• Number of white marbles = 30
• Number of blue marbles = 20
• Number of orange marbles = 15

Summing these quantities gives us the total number of marbles: $\text{Total number of marbles} = 10 + 30 + 20 + 15 = 75$ So, there are 75 marbles in the box.

Step 2: Calculate the Probability of the First Event (Drawing a Red Marble) Let event A be "the first marble drawn is red." We apply the probability formula using the number of red marbles and the total number of marbles.

• Favorable outcomes: There are 10 red marbles.
• Total possible outcomes: There are 75 marbles.

The probability of drawing a red marble first is: $P(A) = P(\text{First is red}) = \frac{\text{Number of red marbles}}{\text{Total number of marbles}} = \frac{10}{75}$ To simplify, both 10 and 75 are divisible by 5: $P(A) = \frac{10 \div 5}{75 \div 5} = \frac{2}{15}$

Step 3: Calculate the Probability of the Second Event (Drawing a White Marble) Let event B be "the second marble drawn is white." The problem states "with replacement," which is crucial here. This means that after the first marble was drawn, its color was noted, and then it was put back into the box. Consequently, the composition of the box for the second draw is identical to its original state.

• Favorable outcomes: There are 30 white marbles.
• Total possible outcomes: There are still 75 marbles (due to replacement).

The probability of drawing a white marble second is: $P(B) = P(\text{Second is white}) = \frac{\text{Number of white marbles}}{\text{Total number of marbles}} = \frac{30}{75}$ To simplify, both 30 and 75 are divisible by 15: $P(B) = \frac{30 \div 15}{75 \div 15} = \frac{2}{5}$

Step 4: Calculate the Combined Probability of Both Independent Events Since the draws are made "with replacement," the event of drawing a red marble first and the event of drawing a white marble second are independent. To find the probability that both events occur in this specific order, we multiply their individual probabilities, as per the formula for independent events. $P(\text{First is red and Second is white}) = P(A) \times P(B)$ Substituting the simplified probabilities from Steps 2 and 3: $P(\text{First is red and Second is white}) = \frac{2}{15} \times \frac{2}{5}$ $P(\text{First is red and Second is white}) = \frac{2 \times 2}{15 \times 5} = \frac{4}{75}$

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Common Mistakes & Tips

1. "With Replacement" vs. "Without Replacement": Always carefully read this part of the problem.
   • With Replacement: Events are independent. Probabilities for subsequent draws remain unchanged. Use $P(A \text{ and } B) = P(A) \times P(B)$.
   • Without Replacement: Events are dependent. Probabilities change for subsequent draws because the total number of items and/or the number of specific items decrease. For example, if it were "without replacement," $P(\text{Second is white})$ would be $\frac{30}{74}$ (if the first was red), not $\frac{30}{75}$.
2. Simplifying Fractions: While not strictly necessary until the final answer, simplifying fractions at intermediate steps (e.g., $\frac{10}{75}$ to $\frac{2}{15}$) makes calculations easier and reduces the chance of arithmetic errors.
3. Understanding "And" vs. "Or": In probability, "and" usually implies multiplying probabilities (for independent events), meaning both conditions must be met. "Or" usually implies adding probabilities (for mutually exclusive events), meaning at least one condition must be met.

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Summary

We first determined the total number of marbles. Then, we calculated the probability of drawing a red marble in the first draw. Because the drawing is done "with replacement," the conditions for the second draw are identical to the first. We then calculated the probability of drawing a white marble in the second draw. Since these events are independent, the overall probability of both occurring in succession is the product of their individual probabilities. This yielded a result of $\frac{4}{75}$.

The final answer is $\boxed{\frac{4}{75}}$ which corresponds to option (D).