Answer :
Let's analyze the problem of finding the condition that must be true if the two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent.
We start with the definition of independent events. Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if and only if the occurrence of one event does not affect the probability of the occurrence of the other event. Mathematically, this means:
[tex]\[ P(A \cap B) = P(A) \cdot P(B) \][/tex]
Now, let's address each answer choice:
### Option A: [tex]\( P(B \mid A) = x y \)[/tex]
The conditional probability [tex]\( P(B \mid A) \)[/tex] is the probability of [tex]\( B \)[/tex] occurring given that [tex]\( A \)[/tex] has already occurred. According to the definition of conditional probability:
[tex]\[ P(B \mid A) = \frac{P(A \cap B)}{P(A)} \][/tex]
For independent events, [tex]\( P(A \cap B) = P(A) \cdot P(B) \)[/tex]. Substituting, we get:
[tex]\[ P(B \mid A) = \frac{P(A) \cdot P(B)}{P(A)} = P(B) \][/tex]
Therefore, [tex]\( P(B \mid A) = y \)[/tex], not [tex]\( x y \)[/tex]. So, option A is incorrect.
### Option B: [tex]\( P(A \mid B) = x \)[/tex]
The conditional probability [tex]\( P(A \mid B) \)[/tex] is the probability of [tex]\( A \)[/tex] occurring given that [tex]\( B \)[/tex] has already occurred. Using the definition of conditional probability:
[tex]\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \][/tex]
For independent events, [tex]\( P(A \cap B) = P(A) \cdot P(B) \)[/tex]. Substituting, we get:
[tex]\[ P(A \mid B) = \frac{P(A) \cdot P(B)}{P(B)} = P(A) \][/tex]
Therefore, [tex]\( P(A \mid B) = x \)[/tex]. So, option B is correct.
### Option C: [tex]\( P(B \mid A) = x \)[/tex]
We have already evaluated [tex]\( P(B \mid A) \)[/tex] above. We found that [tex]\( P(B \mid A) = P(B) = y \)[/tex], not [tex]\( x \)[/tex]. So, option C is incorrect.
### Option D: [tex]\( P(A \mid B) = y \)[/tex]
Again, we have calculated that [tex]\( P(A \mid B) = x \)[/tex]. So, option D is incorrect.
### Conclusion
The correct condition that must be true if events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent is given in option B:
[tex]\[ P(A \mid B) = x \][/tex]
We start with the definition of independent events. Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if and only if the occurrence of one event does not affect the probability of the occurrence of the other event. Mathematically, this means:
[tex]\[ P(A \cap B) = P(A) \cdot P(B) \][/tex]
Now, let's address each answer choice:
### Option A: [tex]\( P(B \mid A) = x y \)[/tex]
The conditional probability [tex]\( P(B \mid A) \)[/tex] is the probability of [tex]\( B \)[/tex] occurring given that [tex]\( A \)[/tex] has already occurred. According to the definition of conditional probability:
[tex]\[ P(B \mid A) = \frac{P(A \cap B)}{P(A)} \][/tex]
For independent events, [tex]\( P(A \cap B) = P(A) \cdot P(B) \)[/tex]. Substituting, we get:
[tex]\[ P(B \mid A) = \frac{P(A) \cdot P(B)}{P(A)} = P(B) \][/tex]
Therefore, [tex]\( P(B \mid A) = y \)[/tex], not [tex]\( x y \)[/tex]. So, option A is incorrect.
### Option B: [tex]\( P(A \mid B) = x \)[/tex]
The conditional probability [tex]\( P(A \mid B) \)[/tex] is the probability of [tex]\( A \)[/tex] occurring given that [tex]\( B \)[/tex] has already occurred. Using the definition of conditional probability:
[tex]\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \][/tex]
For independent events, [tex]\( P(A \cap B) = P(A) \cdot P(B) \)[/tex]. Substituting, we get:
[tex]\[ P(A \mid B) = \frac{P(A) \cdot P(B)}{P(B)} = P(A) \][/tex]
Therefore, [tex]\( P(A \mid B) = x \)[/tex]. So, option B is correct.
### Option C: [tex]\( P(B \mid A) = x \)[/tex]
We have already evaluated [tex]\( P(B \mid A) \)[/tex] above. We found that [tex]\( P(B \mid A) = P(B) = y \)[/tex], not [tex]\( x \)[/tex]. So, option C is incorrect.
### Option D: [tex]\( P(A \mid B) = y \)[/tex]
Again, we have calculated that [tex]\( P(A \mid B) = x \)[/tex]. So, option D is incorrect.
### Conclusion
The correct condition that must be true if events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent is given in option B:
[tex]\[ P(A \mid B) = x \][/tex]
