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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]$P(A \mid B)=P(B)$[/tex].

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

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

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


Sagot :

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].