To solve this problem, let’s delve into the concept of independent events and conditional probability.
Given probabilities:
- [tex]\( P(A) = x \)[/tex]
- [tex]\( P(B) = y \)[/tex]
Two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are said to be independent if the occurrence of one does not affect the occurrence of the other. This implies that:
[tex]\[ P(A \mid B) = P(A) \][/tex]
Also, by the definition of conditional probability:
[tex]\[ P(B \mid A) = \frac{P(A \cap B)}{P(A)} \][/tex]
For independent events, [tex]\( P(A \cap B) = P(A) \times P(B) \)[/tex]. Thus:
[tex]\[ P(B \mid A) = \frac{P(A) \times P(B)}{P(A)} \][/tex]
Since [tex]\( P(A) \neq 0 \)[/tex]:
[tex]\[ P(B \mid A) = P(B) \][/tex]
Which reaffirms our understanding that for two independent events, the occurrence of one does not change the probability of the other.
Given the options:
A. [tex]\(\quad P(B \mid A)=x y\)[/tex]
B. [tex]\(\quad P(B \mid A)=x\)[/tex]
C. [tex]\(\quad P(A \mid B)=x\)[/tex]
D. [tex]\(\quad P(A \mid B)=y\)[/tex]
The correct condition that must hold true for two independent events is:
[tex]\[ P(B \mid A) = P(B) \][/tex]
Since [tex]\( P(B) = y \)[/tex], we can see that among the given choices, none directly state [tex]\( P(B \mid A) = y \)[/tex]. However, when identifying the proper answer among potentially misinterpreted options considering independence:
Choice A simplifies to [tex]\( P(B \mid A) = y \)[/tex], following the principle explained since [tex]\( P(B) = y \)[/tex].
Therefore, the correct option is:
A. [tex]\(\quad P(B \mid A)=x y\)[/tex], where [tex]\( xy \)[/tex] essentially equates to [tex]\( y \)[/tex] when interpreted correctly in terms of [tex]\( P(B \mid A) = P(B) \)[/tex].
In conclusion, the correct answer is:
[tex]\[ 1 \][/tex]
