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

Sagot :

GinnyAnswer
Sure! Let's analyze the situation step by step.

We have the following probabilities given in the problem:

1. The probability that Edward purchases a video game from the store is [tex]\( P(A) = 0.67 \)[/tex].
2. The probability that Greg purchases a video game from the store is [tex]\( P(B) = 0.74 \)[/tex].
3. The probability that Edward purchases a video game given that Greg has purchased a video game is [tex]\( P(A \mid B) = 0.67 \)[/tex].

Now, we need to determine whether events [tex]\( A \)[/tex] (Edward purchasing a video game) and [tex]\( B \)[/tex] (Greg purchasing a video game) are independent or dependent.

To check for independence, we use the definition of independent events. Two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if and only if:
[tex]$ P(A \mid B) = P(A) $[/tex]

Given:
[tex]$ P(A \mid B) = 0.67 $[/tex]
[tex]$ P(A) = 0.67 $[/tex]

Since [tex]\( P(A \mid B) = P(A) \)[/tex], we can conclude that Edward purchasing a video game is independent of Greg purchasing a video game.

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].

Explanation of other incorrect options:
- Option B is incorrect because it states [tex]\( P(A \mid B) \neq P(A) \)[/tex], which is not true given the values.
- Option C is also incorrect because independence is defined by [tex]\( P(A \mid B) = P(A) \)[/tex].
- Option D is incorrect because independence is defined by [tex]\( P(A \mid B) = P(A) \)[/tex], not [tex]\( P(A \mid B) = P(B) \)[/tex].

Thus, the correct answer is A.
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