To determine the correct condition for independent events, let's recall the definition of independent events in probability.
For two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] to be independent, the occurrence of one event must not affect the probability of the other. Mathematically, this is expressed as:
[tex]\[ P(A \cap B) = P(A) \cdot P(B) \][/tex]
Alternatively, given that [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent, it also follows that:
[tex]\[ P(B \mid A) = P(B) \][/tex]
and
[tex]\[ P(A \mid B) = P(A) \][/tex]
Now, let's evaluate the given options:
A. [tex]\( P(B \mid A) = xy \)[/tex]
- This is not correct as the conditional probability [tex]\( P(B \mid A) \)[/tex] should equal [tex]\( P(B) \)[/tex], not [tex]\( x \cdot y \)[/tex].
B. [tex]\( P(B \mid A) = x \)[/tex]
- This is incorrect because for independent events, [tex]\( P(B \mid A) \)[/tex] equals [tex]\( P(B) \)[/tex], not [tex]\( P(A) \)[/tex].
C. [tex]\( P(A \mid B) = x \)[/tex]
- This option suggests that the conditional probability of [tex]\( A \)[/tex] given [tex]\( B \)[/tex] equals the probability of [tex]\( A \)[/tex] itself, which is correct for independent events: [tex]\( P(A \mid B) = P(A) \)[/tex].
D. [tex]\( P(A \mid B) = y \)[/tex]
- This is not correct since for independent events, [tex]\( P(A \mid B) \)[/tex] should equal [tex]\( P(A) \)[/tex], and not [tex]\( P(B) \)[/tex].
Therefore, the correct condition for independent events is given by:
[tex]\[ \boxed{C} \: P(A \mid B) = x \][/tex]
This reflects the property that for independent events, the conditional probability of [tex]\( A \)[/tex] given [tex]\( B \)[/tex] is simply the probability of [tex]\( A \)[/tex] itself.
