To determine the correct condition that must be true when events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent, let’s start by revisiting the definition of independent events in probability.
### Definition of Independent Events
Two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are said to be independent if the occurrence of one event does not affect the probability of the other event occurring. Mathematically, this is expressed as:
[tex]\[ P(A \cap B) = P(A) \cdot P(B) \][/tex]
Additionally, for independent events [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ P(A \mid B) = P(A) \][/tex]
[tex]\[ P(B \mid A) = P(B) \][/tex]
### Analyzing Each Option
Let’s analyze each option given in the question:
A. [tex]\( P(A \mid B) = x \)[/tex]
Since [tex]\( x \)[/tex] represents [tex]\( P(A) \)[/tex] and knowing that [tex]\( P(A \mid B) = P(A) \)[/tex], we can rewrite this as:
[tex]\[ P(A \mid B) = P(A) \][/tex]
Thus, this condition simplifies to:
[tex]\[ P(A \mid B) = x \][/tex]
This condition is true for independent events.
B. [tex]\( P(A \mid B) = y \)[/tex]
Here, [tex]\( y \)[/tex] represents [tex]\( P(B) \)[/tex]. For independent events, [tex]\( P(A \mid B) \)[/tex] should equal [tex]\( P(A) \)[/tex] and not [tex]\( P(B) \)[/tex]. Therefore:
[tex]\[ P(A \mid B) \neq P(B) \][/tex]
This condition is false for independent events.
C. [tex]\( P(B \mid A) = x \)[/tex]
In this case, [tex]\( x \)[/tex] still represents [tex]\( P(A) \)[/tex]. For independent events, [tex]\( P(B \mid A) \)[/tex] should equal [tex]\( P(B) \)[/tex] and not [tex]\( P(A) \)[/tex]. Thus:
[tex]\[ P(B \mid A) \neq P(A) \][/tex]
This condition is false for independent events.
D. [tex]\( P(B \mid A) = xy \)[/tex]
This implies that [tex]\( P(B \mid A) \)[/tex] is equal to the product of [tex]\( P(A) \)[/tex] and [tex]\( P(B) \)[/tex]. For independent events, we know:
[tex]\[ P(B \mid A) = P(B) \][/tex]
Therefore, equating it to [tex]\( xy \)[/tex] would not be correct:
[tex]\[ P(B \mid A) \neq P(A) \cdot P(B) \][/tex]
This condition is false for independent events.
### Conclusion
Based on the analysis above, the correct condition that must be true when events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent is:
A. [tex]\( P(A \mid B) = x \)[/tex]
