Sure! Let's analyze the situation step by step.
We have the following probabilities given in the problem:
1. The probability that Edward purchases a video game from the store is [tex]\( P(A) = 0.67 \)[/tex].
2. The probability that Greg purchases a video game from the store is [tex]\( P(B) = 0.74 \)[/tex].
3. The probability that Edward purchases a video game given that Greg has purchased a video game is [tex]\( P(A \mid B) = 0.67 \)[/tex].
Now, we need to determine whether events [tex]\( A \)[/tex] (Edward purchasing a video game) and [tex]\( B \)[/tex] (Greg purchasing a video game) are independent or dependent.
To check for independence, we use the definition of independent events. Two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if and only if:
[tex]$ P(A \mid B) = P(A) $[/tex]
Given:
[tex]$ P(A \mid B) = 0.67 $[/tex]
[tex]$ P(A) = 0.67 $[/tex]
Since [tex]\( P(A \mid B) = P(A) \)[/tex], we can conclude that Edward purchasing a video game is independent of Greg purchasing a video game.
Therefore, the correct statement is:
A. Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent because [tex]\( P(A \mid B) = P(A) \)[/tex].
Explanation of other incorrect options:
- Option B is incorrect because it states [tex]\( P(A \mid B) \neq P(A) \)[/tex], which is not true given the values.
- Option C is also incorrect because independence is defined by [tex]\( P(A \mid B) = P(A) \)[/tex].
- Option D is incorrect because independence is defined by [tex]\( P(A \mid B) = P(A) \)[/tex], not [tex]\( P(A \mid B) = P(B) \)[/tex].
Thus, the correct answer is A.
