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



Answer :

GinnyAnswer
To solve this problem, let's understand the relationships between events [tex]\( A \)[/tex] and [tex]\( B \)[/tex].

1. Definition of Conditional Probability:
- [tex]\( P(A \mid B) \)[/tex] represents the probability of event [tex]\( A \)[/tex] occurring given that event [tex]\( B \)[/tex] has occurred.

2. Given Data:
- [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 purchases a video game.

3. Independence of Events:
- Two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are considered independent if the occurrence of one does not affect the probability of the other.
- Mathematically, [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if and only if [tex]\( P(A \mid B) = P(A) \)[/tex].

Now let's verify the independence of events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] using the given data.
- We compare [tex]\( P(A \mid B) \)[/tex] with [tex]\( P(A) \)[/tex].
- Given [tex]\( P(A \mid B) = 0.67 \)[/tex] and [tex]\( P(A) = 0.67 \)[/tex].

Since [tex]\( P(A \mid B) = P(A) \)[/tex], this indicates that the occurrence of event [tex]\( B \)[/tex] (Greg purchasing a video game) does not change the probability of event [tex]\( A \)[/tex] (Edward purchasing a video game).

Therefore, events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent.

Among the given options:
- A. Incorrect: The statement mixes the probabilities. [tex]\( P(A \mid B) = P(B) \)[/tex] is not relevant in this context.
- B. Incorrect: This would be true if [tex]\( P(A \mid B) \neq P(A) \)[/tex], but here they are equal.
- C. Correct: This matches our conclusion that events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent because [tex]\( P(A \mid B) = P(A) \)[/tex].
- D. Incorrect: This statement incorrectly declares dependence even though [tex]\( P(A \mid B) = P(A) \)[/tex], which actually implies independence.

Therefore, the correct statement is:
- C. Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent because [tex]\( P(A \mid B) = P(A) \)[/tex].