To solve this problem, we need to determine the relationship between two events, [tex]\( A \)[/tex] (Edward purchasing a video game) and [tex]\( B \)[/tex] (Greg purchasing a video game), based on given probabilities.
We are given the following probabilities:
- [tex]\( P(A) = 0.67 \)[/tex]: The probability that Edward purchases a video game.
- [tex]\( P(B) = 0.74 \)[/tex]: The probability that Greg purchases a video game.
- [tex]\( P(A \mid B) = 0.67 \)[/tex]: The probability that Edward purchases a video game given that Greg has purchased one.
To determine whether events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent, we need to check the condition for independence of events:
[tex]\[ P(A \mid B) = P(A) \][/tex]
Independence of events means that the occurrence of one event does not affect the probability of the other event. Hence if events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent, then:
[tex]\[ P(A \mid B) = P(A) \][/tex]
From the given probabilities, [tex]\( P(A \mid B) = 0.67 \)[/tex] and [tex]\( P(A) = 0.67 \)[/tex].
Since:
[tex]\[ P(A \mid B) = P(A) \][/tex]
[tex]\[ 0.67 = 0.67 \][/tex]
This condition is satisfied, which means that events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent.
Therefore, the correct statement is:
B. Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent because [tex]\( P(A \mid B) = P(A) \)[/tex].
