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Select the correct answer.

The probability that Edward purchases a video game from a store is 0.67 (event [tex]$A$[/tex]), and the probability that Greg purchases a video game from the store is 0.74 (event [tex]$B$[/tex]). The probability that Edward purchases a video game given that Greg has purchased a video game is 0.67.

Which statement is true?

A. Events [tex]$A$[/tex] and [tex]$B$[/tex] are dependent because [tex]$P(A \mid B)=P(A)$[/tex].
B. Events [tex]$A$[/tex] and [tex]$B$[/tex] are dependent because [tex]$P(A \mid B) \neq P(A)$[/tex].
C. Events [tex]$A$[/tex] and [tex]$B$[/tex] are independent because [tex]$P(A \mid B)=P(A)$[/tex].
D. Events [tex]$A$[/tex] and [tex]$B$[/tex] are independent because [tex]$P(A \mid B)=P(B)$[/tex].



Answer :

GinnyAnswer
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]

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