To solve this problem, let's review the concept of independence in probability:
When two events \( A \) and \( B \) are independent, the occurrence of one event does not affect the probability of the other. Mathematically, this means:
[tex]\[ P(A \cap B) = P(A) \times P(B) \][/tex]
Also, the conditional probabilities for independent events can be expressed as follows:
[tex]\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{P(A) \times P(B)}{P(B)} = P(A) \][/tex]
[tex]\[ P(B \mid A) = \frac{P(B \cap A)}{P(A)} = \frac{P(A) \times P(B)}{P(A)} = P(B) \][/tex]
Given that:
- \( P(A) = x \)
- \( P(B) = y \)
We can rewrite the conditional probabilities:
[tex]\[ P(A \mid B) = x \][/tex]
[tex]\[ P(B \mid A) = y \][/tex]
With this understanding, let's analyze the given choices:
A. \( P(A \mid B) = y \)
- This suggests that the conditional probability of \( A \) given \( B \) is \( y \), but for independent events, \( P(A \mid B) \) should equal \( P(A) = x \), so this statement is incorrect.
B. \( P(B \mid A) = x \)
- This suggests that the conditional probability of \( B \) given \( A \) is \( x \), but for independent events, \( P(B \mid A) \) should equal \( P(B) = y \), so this statement is incorrect.
C. \( P(B \mid A) = xy \)
- This suggests that the conditional probability of \( B \) given \( A \) is \( x \times y \), which does not match our findings for independent events, so this statement is incorrect.
D. \( P(A \mid B) = x \)
- This suggests that the conditional probability of \( A \) given \( B \) is \( x \), which aligns perfectly with our understanding that \( P(A \mid B) = P(A) = x \) for independent events, so this statement is correct.
Therefore, the correct answer is:
D. [tex]\( P(A \mid B) = x \)[/tex]
