To solve this question, 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 occurrence of the other event. This means:
[tex]\[ P(A \mid B) = P(A) \][/tex]
Given:
- The probability of event [tex]\( A \)[/tex] is [tex]\( x \)[/tex].
- The probability of event [tex]\( B \)[/tex] is [tex]\( y \)[/tex].
We are asked to determine which condition must be true given that the two events are independent.
From the definition of independent events, we know that:
[tex]\[ P(A \mid B) = P(A) \][/tex]
This is because if event [tex]\( A \)[/tex] and event [tex]\( B \)[/tex] are independent, the probability of [tex]\( A \)[/tex] occurring given that [tex]\( B \)[/tex] has occurred (i.e., [tex]\( P(A \mid B) \)[/tex]), is simply the probability of [tex]\( A \)[/tex] occurring without any consideration of [tex]\( B \)[/tex].
Therefore, the correct condition is:
[tex]\[ P(A \mid B) = P(A) \][/tex]
Since [tex]\( P(A) = x \)[/tex], it follows that:
[tex]\[ P(A \mid B) = x \][/tex]
The correct answer is option A:
[tex]\[ P(A \mid B) = x \][/tex]
