To determine the probability that event \( B \) occurs given that event \( A \) has already occurred, we use the concept of conditional probability. The conditional probability of \( B \) given \( A \) is denoted as \( P(B \mid A) \).
By definition, the conditional probability \( P(B \mid A) \) is given by:
[tex]\[ P(B \mid A) = \frac{P(B \cap A)}{P(A)} \][/tex]
where:
- \( P(B \cap A) \) is the probability that both events \( B \) and \( A \) occur,
- \( P(A) \) is the probability that event \( A \) occurs.
This formula arises from the fundamental principle that conditional probability is the proportion of the probability of the intersection of the two events to the probability of the conditioning event.
Given the options:
A. \( \frac{ P (B \cap A)}{ P (A) \cdot P (B)} \)
B. \( \frac{ P (B \cap A)}{ P (A)} \)
C. \( \frac{ P (B \cap A)}{ P (B)} \)
D. \( \frac{ P (B \cup A)}{ P (B)} \)
We can see that option B corresponds to the correct definition of conditional probability:
[tex]\[ \frac{P(B \cap A)}{P(A)} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
