To determine the correct answer, let's recall the definition of conditional probability. Conditional probability allows us to find the probability of event [tex]\( B \)[/tex] occurring given that event [tex]\( A \)[/tex] has already occurred. This is denoted as [tex]\( P(B|A) \)[/tex].
According to the formula for conditional probability, we have:
[tex]\[
P(B|A) = \frac{P(B \cap A)}{P(A)}
\][/tex]
This formula can be broken down as follows:
- [tex]\( P(B \cap A) \)[/tex] is the probability that both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occur.
- [tex]\( P(A) \)[/tex] is the probability that event [tex]\( A \)[/tex] occurs.
- [tex]\( P(B|A) \)[/tex] is the probability that event [tex]\( B \)[/tex] occurs given that event [tex]\( A \)[/tex] has occurred.
From the options provided:
A. [tex]\( \frac{ P(B \cap A)}{ P(A) \cdot P(B)} \)[/tex]
B. [tex]\( \frac{P(B \cap A)}{P(A)} \)[/tex]
C. [tex]\( \frac{ P(B \cap A)}{P(B)} \)[/tex]
D. [tex]\( \frac{ P(B \cup A)}{ P(B)} \)[/tex]
Based on the definition and formula for conditional probability, the correct answer is:
[tex]\[
B. \frac{P(B \cap A)}{P(A)}
\][/tex]
Hence, the correct answer is:
[tex]\( \boxed{B} \)[/tex]
