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A bag contains a variety of different-colored marbles. If [tex] P(\text{red}) = \frac{1}{2}[/tex], [tex] P(\text{green}) = \frac{1}{4}[/tex], and [tex] P(\text{red and green}) = \frac{1}{8}[/tex], which statement is true?

A. The events are independent because [tex] P(\text{red}) \cdot P(\text{green}) = P(\text{red and green})[/tex].
B. The events are independent because [tex] P(\text{red}) + P(\text{green}) = P(\text{red and green})[/tex].
C. The events are dependent because [tex] P(\text{red}) \cdot P(\text{green}) \neq P(\text{red and green})[/tex].
D. The events are dependent because [tex] P(\text{red}) + P(\text{green}) \neq P(\text{red and green})[/tex].

Sagot :

Let's analyze the problem step by step.

1. We are given three probabilities:
- [tex]\(P(\text{red}) = \frac{1}{2}\)[/tex]
- [tex]\(P(\text{green}) = \frac{1}{4}\)[/tex]
- [tex]\(P(\text{red and green}) = \frac{1}{8}\)[/tex]

2. We need to determine if the events "red" and "green" are independent or dependent. Events are independent if the probability of both occurring together [tex]\(P(\text{red and green})\)[/tex] is equal to the product of their individual probabilities [tex]\(P(\text{red}) \cdot P(\text{green})\)[/tex].

3. First, let's calculate:
- The product of [tex]\(P(\text{red})\)[/tex] and [tex]\(P(\text{green})\)[/tex]:
[tex]\[ P(\text{red}) \cdot P(\text{green}) = \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8} \][/tex]

- The sum of [tex]\(P(\text{red})\)[/tex] and [tex]\(P(\text{green})\)[/tex]:
[tex]\[ P(\text{red}) + P(\text{green}) = \frac{1}{2} + \frac{1}{4} = \frac{3}{4} \][/tex]

4. Now, compare these values with [tex]\(P(\text{red and green})\)[/tex]:
- For the product:
[tex]\[ P(\text{red}) \cdot P(\text{green}) = \frac{1}{8} \][/tex]
This is equal to [tex]\(P(\text{red and green})\)[/tex].

- For the sum:
[tex]\[ P(\text{red}) + P(\text{green}) = \frac{3}{4} \][/tex]
This is not equal to [tex]\(P(\text{red and green})\)[/tex].

5. Since the product [tex]\(P(\text{red}) \cdot P(\text{green})\)[/tex] is equal to [tex]\(P(\text{red and green})\)[/tex], the events "red" and "green" are independent.

Therefore, the correct statement is:
The events are independent because [tex]\(P(\text{red}) \cdot P(\text{green}) = P(\text{red and green})\)[/tex].