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



Answer :

GinnyAnswer
To determine whether events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent or dependent, we need to compare the probability [tex]\( P(A \mid B) \)[/tex] (the probability that Edward purchases a video game given that Greg has purchased a video game) with [tex]\( P(A) \)[/tex] (the probability that Edward purchases a video game).

Given:
- [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 a video game)

Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if and only if:
[tex]\[ P(A \mid B) = P(A) \][/tex]

In this case:
[tex]\[ P(A \mid B) = 0.67 \][/tex]
[tex]\[ P(A) = 0.67 \][/tex]

Since [tex]\( P(A \mid B) \)[/tex] equals [tex]\( P(A) \)[/tex], the events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent.

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]

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