When considering whether two events, [tex]\( A \)[/tex] and [tex]\( B \)[/tex], are independent, we need to examine the relationship between their probabilities.
The definition of independence for two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is:
[tex]\[ P(A \cap B) = P(A) \cdot P(B) \][/tex]
This means the occurrence of event [tex]\( A \)[/tex] does not affect the probability of occurrence of event [tex]\( B \)[/tex], and vice versa.
We can also express this relationship in terms of conditional probabilities. If two events are independent, 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]. In other words:
[tex]\[ P(A \mid B) = P(A) \][/tex]
This is because the occurrence of [tex]\( B \)[/tex] does not influence the occurrence of [tex]\( A \)[/tex] if they are independent.
To break it down step-by-step:
1. Given:
- Two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent.
2. Definition of independence:
- By definition, [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if [tex]\( P(A \cap B) = P(A) \cdot P(B) \)[/tex].
3. Conditional probability:
- Conditional probability [tex]\( P(A \mid B) \)[/tex] is defined as:
[tex]\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
\][/tex]
4. Substitute the independence condition:
- Since [tex]\( P(A \cap B) = P(A) \cdot P(B) \)[/tex], we substitute this into the conditional probability formula:
[tex]\[
P(A \mid B) = \frac{P(A) \cdot P(B)}{P(B)}
\][/tex]
5. Simplify the expression:
- Simplifying the right-hand side, we get:
[tex]\[
P(A \mid B) = P(A)
\][/tex]
Therefore, the correct statement when events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent is:
[tex]\[ P(A \mid B) = P(A) \][/tex]
So, examining the options:
- [tex]\( P(A \mid B )=P(B) \)[/tex] is incorrect because it does not define the independence of [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
- [tex]\( P(A \mid B )=P(A) \)[/tex] is correct as it matches our derived condition for independence.
- [tex]\( P(A)=P(B) \)[/tex] is incorrect as it assumes both events have the same probability, which is not necessarily the case.
- [tex]\( P(A \mid B )=P(B \mid A ) \)[/tex] is also incorrect because it does not define the independence of the events directly.
Hence, the correct answer is:
[tex]\[ P(A \mid B )=P(A) \][/tex]
