To find the correct expression equal to the probability of event [tex]\( A \)[/tex], we can utilize the concept of complementary events and the values provided.
Given:
- The complement of the probability of event [tex]\( A \)[/tex] is [tex]\( 0.6 \)[/tex].
We know that the probability of an event [tex]\( A \)[/tex] plus the probability of its complement [tex]\( \overline{A} \)[/tex] equals 1. Therefore, the probability of event [tex]\( A \)[/tex] can be calculated as:
[tex]\[
P(A) = 1 - P(\overline{A})
\][/tex]
Using the given information:
[tex]\[
P(A) = 1 - 0.6 = 0.4
\][/tex]
Now let's evaluate the provided options to find which one equals [tex]\( 0.4 \)[/tex]:
A. [tex]\(\frac{P(B \cap A)}{P(B)}\)[/tex]
B. [tex]\(\frac{P(B \cap A)}{P(B \mid A)}\)[/tex]
C. [tex]\(\frac{P(A \cap B)}{P(A \mid B)}\)[/tex]
D. [tex]\(P(A \cap B) \times P(B \mid A)\)[/tex]
E. [tex]\(P(A \cap B) \times P(B)\)[/tex]
Note that [tex]\( \frac{P(B \cap A)}{P(B)} \)[/tex] and [tex]\( \frac{P(A \cap B)}{P(A \mid B)} \)[/tex] and so on do not directly express the probability of [tex]\( A \)[/tex] which we derived as [tex]\( 0.4 \)[/tex].
After careful examination, the correct option that directly addresses the probability [tex]\( 0.4 \)[/tex] is option:
B. [tex]\(\frac{P(B \cap A)}{P(B \mid A)}\)[/tex]
This expression simplifies to [tex]\( P(A) \)[/tex].
Thus, the correct answer is (B).
