Let's work through this step by step to determine the condition that must be true if two events, [tex]\( A \)[/tex] and [tex]\( B \)[/tex], are independent.
1. Understand the definition of independent events:
- Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if the occurrence of one does not affect the probability of the occurrence of the other.
- Mathematically, this is stated as [tex]\( P(A \mid B) = P(A) \)[/tex]. This means that the probability of event [tex]\( A \)[/tex] occurring given that [tex]\( B \)[/tex] has occurred is just the probability of [tex]\( A \)[/tex] occurring, irrespective of [tex]\( B \)[/tex].
2. Given probabilities:
- [tex]\( P(A) = x \)[/tex]
- [tex]\( P(B) = y \)[/tex]
3. Analyze each option:
- Option A: [tex]\( P(A \mid B) = x \)[/tex]
- This directly states that [tex]\( P(A \mid B) = P(A) \)[/tex].
- Since [tex]\( P(A) = x \)[/tex], this condition means [tex]\( P(A \mid B) = x \)[/tex], which is the definition of independence.
- Option B: [tex]\( P(A \mid B) = y \)[/tex]
- This is incorrect because [tex]\( P(A \mid B) \)[/tex] should be equal to [tex]\( P(A) \)[/tex], not [tex]\( P(B) \)[/tex]. There is no reason for [tex]\( P(A \mid B) \)[/tex] to be equal to [tex]\( y \)[/tex].
- Option C: [tex]\( P(B \mid A) = x \)[/tex]
- This is incorrect because [tex]\( P(B \mid A) \)[/tex] should be equal to [tex]\( P(B) \)[/tex] if [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent. It should not be equal to [tex]\( x \)[/tex], which is [tex]\( P(A) \)[/tex].
- Option D: [tex]\( P(B \mid A) = xy \)[/tex]
- This is also incorrect. For independent events, [tex]\( P(B \mid A) = P(B) \)[/tex], not [tex]\( P(B) \)[/tex] times [tex]\( P(A) \)[/tex].
4. Conclusion:
- The correct answer is Option A: [tex]\( P(A \mid B) = x \)[/tex], because it reflects the definition of independent events: [tex]\( P(A \mid B) = P(A) \)[/tex].
Therefore, the condition that must be true when two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent is:
[tex]\[ \boxed{1} \][/tex]
which corresponds to [tex]\( P(A \mid B) = x \)[/tex].
