To find the probability of event [tex]\( A \)[/tex] when event [tex]\( B \)[/tex] is dependent on event [tex]\( A \)[/tex] and event [tex]\( A \)[/tex] occurs before event [tex]\( B \)[/tex], we use the concept of conditional probability.
Given that [tex]\( B \)[/tex] depends on [tex]\( A \)[/tex], the relationship between their probabilities can be expressed using the definition of conditional probability:
[tex]\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \][/tex]
However, we need the expression for [tex]\( P(A) \)[/tex]. Based on the information provided, let’s examine each option to determine the correct one.
A. [tex]\[ \frac{P(B \cap A)}{P(B)} \][/tex]
This expression represents the conditional probability [tex]\( P(A \mid B) \)[/tex], not [tex]\( P(A) \)[/tex]. Hence, this option is incorrect.
B. [tex]\[ \frac{P(B \cap A)}{P(B \mid A)} \][/tex]
This expression rearranges to fit the conditional probability formula for [tex]\( P(A) \)[/tex]. Specifically:
[tex]\[ P(A) = \frac{P(B \cap A)}{P(B \mid A)} \][/tex]
This accurately represents [tex]\( P(A) \)[/tex] when event [tex]\( B \)[/tex] is dependent on event [tex]\( A \)[/tex]. So, this option is correct.
C. [tex]\[ \frac{P(A \cap B)}{P(A \mid B)} \][/tex]
This expression rearranges to:
[tex]\[ P(A \mid B) = \frac{P(A \cap B)}{P(A)} \][/tex]
However, we need [tex]\( P(A) \)[/tex] and this expression assumes an incorrect order of dependence. Thus, this option is incorrect.
D. [tex]\[ P(A \cap B) \times P(B \mid A) \][/tex]
This expression represents the joint probability formula and is not equal to [tex]\( P(A) \)[/tex]. Therefore, this option is incorrect.
E. [tex]\[ P(A \cap B) \times P(B) \][/tex]
This is a different form of joint probability assuming independence, which is not given in the problem that [tex]\( B \)[/tex] depends on [tex]\( A \)[/tex]. Hence, this option is incorrect.
Based on the analysis, the correct answer is:
[tex]\[ B. \frac{P(B \cap A)}{P(B \mid A)} \][/tex]
