To find [tex]\( P(A \cap B) \)[/tex], which is the probability that both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occur simultaneously, we use the conditional probability formula. This is given by:
[tex]\[ P(A \cap B) = P(A) \cdot P(B \mid A) \][/tex]
Here, we are given:
- [tex]\( P(A) = 0.3 \)[/tex]
- [tex]\( P(B \mid A) = 0.9 \)[/tex]
Substituting these values into the formula:
[tex]\[ P(A \cap B) = 0.3 \cdot 0.9 \][/tex]
[tex]\[ P(A \cap B) = 0.27 \][/tex]
So, the probability that both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occur is [tex]\( 0.27 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{0.27} \][/tex]
