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].
