To determine the relationship between events \( A \) and \( B \), we need to consider the given probabilities and the definition of independent events.
- We are given that the probability that Edward purchases a video game, \( P(A) \), is 0.67.
- The probability that Greg purchases a video game, \( P(B) \), is 0.74.
- The conditional probability that Edward purchases a video game given that Greg has purchased a video game, \( P(A \mid B) \), is also given as 0.67.
To determine whether events \( A \) and \( B \) are independent or dependent, we should use the definition of independent events:
- Events \( A \) and \( B \) are independent if and only if \( P(A \mid B) = P(A) \).
Here, we see that:
[tex]\[ P(A \mid B) = 0.67 \][/tex]
[tex]\[ P(A) = 0.67 \][/tex]
Since:
[tex]\[ P(A \mid B) = P(A) \][/tex]
This equality demonstrates that event \( A \) (Edward purchasing a video game) is not influenced by event \( B \) (Greg purchasing a video game). Therefore, events \( A \) and \( B \) are independent.
The correct statement is:
C. Events \( A \) and \( B \) are independent because \( P(A \mid B) = P(A) \).
Thus, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
