To determine what must be true if events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent, let's review the definition of independent events in probability theory.
Two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if the occurrence of one event does not affect the occurrence of the other. In mathematical terms, this means that the probability of [tex]\( A \)[/tex] occurring given that [tex]\( B \)[/tex] has occurred is the same as the probability of [tex]\( A \)[/tex] occurring regardless of [tex]\( B \)[/tex]. This is expressed by the equation:
[tex]\[ P(A \mid B) = P(A) \][/tex]
Given this definition, let's examine each of the provided options:
1. [tex]\( P(A \mid B) = P(B) \)[/tex]
- This option suggests that the probability of [tex]\( A \)[/tex] occurring given [tex]\( B \)[/tex] is equal to the probability of [tex]\( B \)[/tex] itself, which is not a necessity for independence. Therefore, this option is incorrect.
2. [tex]\( P(A \mid B) = P(A) \)[/tex]
- This option aligns perfectly with the definition of independence. It states that the probability of [tex]\( A \)[/tex] occurring given [tex]\( B \)[/tex] is equal to the probability of [tex]\( A \)[/tex] occurring on its own, which is precisely the condition for independence. Therefore, this option is correct.
3. [tex]\( P(A) = P(B) \)[/tex]
- This option suggests that the probabilities of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occurring are equal. While this could be true in some cases, it is not a requirement for the events to be independent. Therefore, this option is incorrect.
4. [tex]\( P(A \mid B) = P(B \mid A) \)[/tex]
- This option implies a symmetry between [tex]\( A \)[/tex] and [tex]\( B \)[/tex], but it does not reflect the independence condition directly. Hence, this option is incorrect.
Based on the definitions and analysis, the correct condition for events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] to be independent is:
[tex]\[ P(A \mid B) = P(A) \][/tex]
So, the correct answer is:
[tex]\[ \boxed{P(A \mid B) = P(A)} \][/tex]
