To determine if events [tex]\(A\)[/tex] (Edward purchasing a video game) and [tex]\(B\)[/tex] (Greg purchasing a video game) are independent or dependent, we need to analyze the given probabilities.
### Given probabilities:
- [tex]\(P(A)\)[/tex] = 0.67
- [tex]\(P(B)\)[/tex] = 0.74
- [tex]\(P(A \mid B)\)[/tex] = 0.67
### Definition of Independence:
Events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent if and only if:
[tex]\[ P(A \mid B) = P(A) \][/tex]
Alternatively, this can also be written as:
[tex]\[ P(A \cap B) = P(A) \cdot P(B) \][/tex]
### Analysis:
We are given that:
[tex]\[ P(A \mid B) = 0.67 \][/tex]
According to the definition of independence, for [tex]\(A\)[/tex] and [tex]\(B\)[/tex] to be independent, it must be true that:
[tex]\[ P(A \mid B) = P(A) \][/tex]
From the problem, we have:
[tex]\[ P(A \mid B) = 0.67 \][/tex]
and
[tex]\[ P(A) = 0.67 \][/tex]
Since [tex]\(P(A \mid B) = P(A)\)[/tex], it follows that events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent.
### Conclusion:
Therefore, the correct statement is:
[tex]\[ \text{B: Events A and B are independent because } P(A \mid B) = P(A). \][/tex]
So, the correct answer is:
B. Events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent because [tex]\(P(A \mid B)=P(A)\)[/tex].
