Let's solve this step-by-step.
Given:
1. The probability that Edward purchases a video game, [tex]\( P(A) \)[/tex], is 0.67.
2. The probability that Greg purchases a video game, [tex]\( P(B) \)[/tex], is 0.74.
We need to determine the relationship between these two events based on the conditional probability [tex]\( P(A \mid B) \)[/tex].
### Conditional Probability and Independence
A key point in determining whether two events are independent is to check if the occurrence of one event affects the probability of the other event.
- If events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent, then [tex]\( P(A \mid B) = P(A) \)[/tex].
- If events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are dependent, then [tex]\( P(A \mid B) \neq P(A) \)[/tex].
### Applying the Given Information
From the problem, we know [tex]\( P(A) = 0.67 \)[/tex].
We are given that the conditional probability [tex]\( P(A \mid B) \)[/tex] is equal to [tex]\( P(A) \)[/tex], meaning:
[tex]\[ P(A \mid B) = P(A) = 0.67 \][/tex]
### Analyzing the Statements
1. Option A: Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent because [tex]\( P(A \mid B) = P(A) \)[/tex].
Since [tex]\( P(A \mid B) = P(A) \)[/tex], it confirms that the events are independent.
2. Option B: Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are dependent because [tex]\( P(A \mid B) = P(A) \)[/tex].
This is incorrect since the condition actually shows independence, not dependence.
3. Option C: Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent because [tex]\( P(A \mid B) = P(B) \)[/tex].
This statement is false because independence is defined by [tex]\( P(A \mid B) = P(A) \)[/tex], not [tex]\( P(A \mid B) = P(B) \)[/tex].
4. Option D: Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are dependent because [tex]\( P(A \mid B) \neq P(A) \)[/tex].
This is also incorrect because [tex]\( P(A \mid B) = P(A) \)[/tex].
### Conclusion
The correct statement is:
Option A: Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent because [tex]\( P(A \mid B) = P(A) \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
