Answer :
To determine which condition must be true given that events \(A\) and \(B\) are independent, we need to understand the properties of independent events in probability theory.
Step-by-step Solution:
1. Definition of Independence:
Two events \(A\) and \(B\) are independent if the occurrence of one event does not affect the occurrence of the other. Mathematically, this is expressed as:
[tex]\[ P(A \cap B) = P(A) \cdot P(B) \][/tex]
2. Conditional Probability:
The conditional probability of \(A\) given \(B\) is defined as:
[tex]\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \][/tex]
3. Substitute using Independence:
Since events \(A\) and \(B\) are independent, we can substitute \(P(A \cap B)\) with \(P(A) \cdot P(B)\):
[tex]\[ P(A \mid B) = \frac{P(A) \cdot P(B)}{P(B)} \][/tex]
4. Simplify:
When we simplify the expression, assuming \(P(B) \neq 0\):
[tex]\[ P(A \mid B) = \frac{P(A) \cdot P(B)}{P(B)} = P(A) \][/tex]
5. Conclusion:
Therefore, for independent events \(A\) and \(B\), the conditional probability \(P(A \mid B)\) equals the probability of \(A\), which is \(x\).
Thus, the correct condition is:
[tex]\[ A. \ P(A \mid B)=x \][/tex]
Hence, the correct answer is Option A.
Step-by-step Solution:
1. Definition of Independence:
Two events \(A\) and \(B\) are independent if the occurrence of one event does not affect the occurrence of the other. Mathematically, this is expressed as:
[tex]\[ P(A \cap B) = P(A) \cdot P(B) \][/tex]
2. Conditional Probability:
The conditional probability of \(A\) given \(B\) is defined as:
[tex]\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \][/tex]
3. Substitute using Independence:
Since events \(A\) and \(B\) are independent, we can substitute \(P(A \cap B)\) with \(P(A) \cdot P(B)\):
[tex]\[ P(A \mid B) = \frac{P(A) \cdot P(B)}{P(B)} \][/tex]
4. Simplify:
When we simplify the expression, assuming \(P(B) \neq 0\):
[tex]\[ P(A \mid B) = \frac{P(A) \cdot P(B)}{P(B)} = P(A) \][/tex]
5. Conclusion:
Therefore, for independent events \(A\) and \(B\), the conditional probability \(P(A \mid B)\) equals the probability of \(A\), which is \(x\).
Thus, the correct condition is:
[tex]\[ A. \ P(A \mid B)=x \][/tex]
Hence, the correct answer is Option A.
