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The probability that Edward purchases a video game from a store is 0.67 (event A), and the probability that Greg purchases a video game from the store is 0.74 (event B). 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 A and B are independent because [tex]\( P(A \mid B) = P(B) \)[/tex]
B. Events A and B are dependent because [tex]\( P(A \mid B) = P(A) \)[/tex]
C. Events A and B are dependent because [tex]\( P(A \mid B) \neq P(A) \)[/tex]
D. Events A and B are independent because [tex]\( P(A \mid B) = P(\sqrt{\text{mom}}) \)[/tex]

Sagot :

GinnyAnswer
To determine if events [tex]\(A\)[/tex] (Edward purchasing a video game) and [tex]\(B\)[/tex] (Greg purchasing a video game) are independent or dependent, we need to analyze the given probabilities.

### Given probabilities:
- [tex]\(P(A)\)[/tex] = 0.67
- [tex]\(P(B)\)[/tex] = 0.74
- [tex]\(P(A \mid B)\)[/tex] = 0.67

### Definition of Independence:
Events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent if and only if:
[tex]\[ P(A \mid B) = P(A) \][/tex]
Alternatively, this can also be written as:
[tex]\[ P(A \cap B) = P(A) \cdot P(B) \][/tex]

### Analysis:
We are given that:
[tex]\[ P(A \mid B) = 0.67 \][/tex]

According to the definition of independence, for [tex]\(A\)[/tex] and [tex]\(B\)[/tex] to be independent, it must be true that:
[tex]\[ P(A \mid B) = P(A) \][/tex]

From the problem, we have:
[tex]\[ P(A \mid B) = 0.67 \][/tex]
and
[tex]\[ P(A) = 0.67 \][/tex]

Since [tex]\(P(A \mid B) = P(A)\)[/tex], it follows that events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent.

### Conclusion:
Therefore, the correct statement is:
[tex]\[ \text{B: Events A and B are independent because } P(A \mid B) = P(A). \][/tex]

So, the correct answer is:

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