To solve this problem, we first need to recall what it means for two events to be independent. Two events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent if the occurrence of one event does not affect the probability of the occurrence of the other event. Mathematically, this can be defined as:
[tex]\[ P(A \text{ and } B) = P(A) \cdot P(B) \][/tex]
Given that we have two independent events, [tex]\(A\)[/tex] and [tex]\(B\)[/tex], with probabilities [tex]\(P(A) = x\)[/tex] and [tex]\(P(B) = y\)[/tex], we need to determine which of the given statements is true.
Let's inspect each of the given options:
### Option A: [tex]\( P(B \mid A) = xy \)[/tex]
The conditional probability [tex]\( P(B \mid A) \)[/tex] is defined as the probability of [tex]\(B\)[/tex] occurring given that [tex]\(A\)[/tex] has occurred. Using the definition of conditional probability:
[tex]\[ P(B \mid A) = \frac{P(A \text{ and } B)}{P(A)} \][/tex]
Since [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent events, we can substitute [tex]\( P(A \text{ and } B) \)[/tex] with [tex]\( P(A) \cdot P(B) \)[/tex]:
[tex]\[ P(B \mid A) = \frac{P(A) \cdot P(B)}{P(A)} = \frac{x \cdot y}{x} \][/tex]
Since [tex]\( x \neq 0 \)[/tex],
[tex]\[ P(B \mid A) = y \][/tex]
This shows that option A, which states [tex]\( P(B \mid A) = xy \)[/tex], is incorrect.
### Option B: [tex]\( P(B \mid A) = x \)[/tex]
From the calculation above, we have shown that:
[tex]\[ P(B \mid A) = y \][/tex]
Thus, option B, which states [tex]\( P(B \mid A) = x \)[/tex], is incorrect.
### Option C: [tex]\( P(A \mid B) = x \)[/tex]
Similarly, the conditional probability [tex]\( P(A \mid B) \)[/tex] is given by:
[tex]\[ P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)} \][/tex]
Using the independence of [tex]\(A\)[/tex] and [tex]\(B\)[/tex],
[tex]\[ P(A \mid B) = \frac{P(A) \cdot P(B)}{P(B)} = \frac{x \cdot y}{y} \][/tex]
Since [tex]\( y \neq 0 \)[/tex],
[tex]\[ P(A \mid B) = x \][/tex]
This shows that option C, [tex]\( P(A \mid B) = x \)[/tex], is correct.
### Option D: [tex]\( P(A \mid B) = y \)[/tex]
From our previous calculation,
[tex]\[ P(A \mid B) = x \][/tex]
So, option D, which states [tex]\( P(A \mid B) = y \)[/tex], is incorrect.
Therefore, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
So, among the given options, the condition that must be true for independent events is represented by option C: [tex]\( P(A \mid B) = x \)[/tex].
