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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 a store is 0.74 (event [tex]$B$[/tex]). The probability that Edward purchases a video game given that Greg has purchased one is [tex]$P(A \mid B)$[/tex]. 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
Let's solve this step-by-step.

Given:

1. The probability that Edward purchases a video game, [tex]\( P(A) \)[/tex], is 0.67.
2. The probability that Greg purchases a video game, [tex]\( P(B) \)[/tex], is 0.74.

We need to determine the relationship between these two events based on the conditional probability [tex]\( P(A \mid B) \)[/tex].

### Conditional Probability and Independence

A key point in determining whether two events are independent is to check if the occurrence of one event affects the probability of the other event.

- If events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent, then [tex]\( P(A \mid B) = P(A) \)[/tex].
- If events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are dependent, then [tex]\( P(A \mid B) \neq P(A) \)[/tex].

### Applying the Given Information

From the problem, we know [tex]\( P(A) = 0.67 \)[/tex].

We are given that the conditional probability [tex]\( P(A \mid B) \)[/tex] is equal to [tex]\( P(A) \)[/tex], meaning:
[tex]\[ P(A \mid B) = P(A) = 0.67 \][/tex]

### Analyzing the Statements

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

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

2. Option B: Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are dependent because [tex]\( P(A \mid B) = P(A) \)[/tex].

This is incorrect since the condition actually shows independence, not dependence.

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

This statement is false because independence is defined by [tex]\( P(A \mid B) = P(A) \)[/tex], not [tex]\( P(A \mid B) = P(B) \)[/tex].

4. Option D: Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are dependent because [tex]\( P(A \mid B) \neq P(A) \)[/tex].

This is also incorrect because [tex]\( P(A \mid B) = P(A) \)[/tex].

### Conclusion

The correct statement is:
Option A: Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent because [tex]\( P(A \mid B) = P(A) \)[/tex].

Thus, the correct answer is:
[tex]\[ \boxed{1} \][/tex]