In probability theory, when two events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent, the occurrence of one event does not affect the probability of the occurrence of the other event. This independence can be expressed mathematically:
[tex]\[ P(A \mid B) = P(A) \][/tex]
This formula states that the probability of [tex]\(A\)[/tex] given that [tex]\(B\)[/tex] has occurred, denoted [tex]\(P(A \mid B)\)[/tex], is equal to the probability of [tex]\(A\)[/tex] because the occurrence of [tex]\(B\)[/tex] does not impact [tex]\(A\)[/tex].
Given:
[tex]\[ P(A) = 0.30 \][/tex]
[tex]\[ P(B) = 0.40 \][/tex]
Since [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent:
[tex]\[ P(A \mid B) = P(A) = 0.30 \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{0.30} \][/tex]
