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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]

Sagot :

GinnyAnsweridn
To determine the correct statement regarding the independence or dependence of events [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we need to understand the concept of conditional probability and independence.

1. Given Information:
- [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.

2. Key Concept:
- Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if and only if [tex]\( P(A \mid B) = P(A) \)[/tex].

3. Analysis:
- We are given that [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 satisfies the condition for independence.

4. Conclusion:
- Therefore, [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent events because the condition [tex]\( P(A \mid B) = P(A) \)[/tex] holds true.

Given the provided information and the proper application of the concept of independence, 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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