To solve this problem, we need to understand the concept of independent events in probability theory.
Two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are said to be independent if the occurrence of one event does not affect the probability of the occurrence of the other event. Mathematically, if [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent events, the following condition holds true:
[tex]\[ P(A \mid B) = P(A) \][/tex]
This means that the probability of event [tex]\( A \)[/tex] occurring given that event [tex]\( B \)[/tex] has occurred is simply the probability of event [tex]\( A \)[/tex] occurring, unaffected by [tex]\( B \)[/tex].
Given this understanding, let's analyze the options provided:
- Option A: [tex]\( P(A \mid B)=y \)[/tex]
- This option suggests that the probability of [tex]\( A \)[/tex] given [tex]\( B \)[/tex] equals the probability of [tex]\( B \)[/tex]. This does not align with the definition of independence, which relates [tex]\( P(A \mid B) \)[/tex] to [tex]\( P(A) \)[/tex], not [tex]\( P(B) \)[/tex].
- Option B: [tex]\( P(B \mid A)=x \)[/tex]
- This option suggests that the probability of [tex]\( B \)[/tex] given [tex]\( A \)[/tex] equals the probability of [tex]\( A \)[/tex]. This does not align with the definition of independence. The correct relation for independent events would involve comparing [tex]\( P(B \mid A) \)[/tex] to [tex]\( P(B) \)[/tex].
- Option C: [tex]\( P(A \mid B)=x \)[/tex]
- This option is in line with the definition of independence. It directly states that [tex]\( P(A \mid B) \)[/tex] equals [tex]\( x \)[/tex], which is the same as [tex]\( P(A) \)[/tex]. This matches the condition for independent events where [tex]\( P(A \mid B) = P(A) \)[/tex].
- Option D: [tex]\( P(B \mid A)=x y \)[/tex]
- This option suggests that [tex]\( P(B \mid A) \)[/tex] is the product of [tex]\( x \)[/tex] and [tex]\( y \)[/tex], which does not relate correctly to the definition of independence. The independence condition requires that [tex]\( P(A \mid B) = P(A) \)[/tex] and [tex]\( P(B \mid A) = P(B) \)[/tex], without introducing the product of [tex]\( P(A) \)[/tex] and [tex]\( P(B) \)[/tex].
Therefore, based on the condition for the independence of events, the correct answer is Option C: [tex]\( P(A \mid B)=x \)[/tex]. This satisfies the requirement that [tex]\( P(A \mid B) = P(A) \)[/tex] for independent events.
