To determine which statement is true, we need to analyze the given probabilities:
1. Probability that Edward purchases a video game (Event [tex]\( A \)[/tex]):
[tex]\[
P(A) = 0.67
\][/tex]
2. Probability that Greg purchases a video game (Event [tex]\( B \)[/tex]):
[tex]\[
P(B) = 0.74
\][/tex]
3. Probability that Edward purchases a video game given that Greg has purchased a video game:
[tex]\[
P(A \mid B) = 0.67
\][/tex]
For two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] to be independent, the condition must hold that:
[tex]\[
P(A \mid B) = P(A)
\][/tex]
Let's examine this condition with the given data:
[tex]\[
P(A \mid B) = 0.67 \quad \text{and} \quad P(A) = 0.67
\][/tex]
Here, [tex]\( P(A \mid B) = P(A) \)[/tex]. This shows that Edward purchasing a video game is independent of whether Greg has purchased a video game.
Let's look at the given statements in the question:
A. Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent because [tex]\( P(A \mid B) = P(B) \)[/tex].
This statement is incorrect because it states the wrong condition for independence. Independence requires [tex]\( P(A \mid B) = P(A) \)[/tex], not [tex]\( P(B) \)[/tex].
B. Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent because [tex]\( P(A \mid B) = P(A) \)[/tex].
This statement is correct as it aligns with our condition for independence. [tex]\( P(A \mid B) = P(A) \)[/tex] suggests that the events are independent.
C. Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are dependent because [tex]\( P(A \mid B) \neq P(A) \)[/tex].
This statement is incorrect because [tex]\( P(A \mid B) = P(A) \)[/tex], suggesting independence, not dependence.
D. Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are dependent because [tex]\( P(A \mid B) = P(A) \)[/tex].
This statement is incorrect because if [tex]\( P(A \mid B) = P(A) \)[/tex], the events are independent, not dependent.
Therefore, the correct answer is:
B. Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent because [tex]\( P(A \mid B)=P(A) \)[/tex].
