To solve for [tex]\( P(A \cap B) \)[/tex], we can use the definition of conditional probability. The formula for conditional probability is given by:
[tex]\[ P(B \mid A) = \frac{P(A \cap B)}{P(A)} \][/tex]
Since we need to find [tex]\( P(A \cap B) \)[/tex], we can rearrange this formula to solve for [tex]\( P(A \cap B) \)[/tex]:
[tex]\[ P(A \cap B) = P(A) \times P(B \mid A) \][/tex]
Given:
[tex]\[ P(A) = 0.4 \][/tex]
[tex]\[ P(B \mid A) = 0.8 \][/tex]
Substitute these values into the formula:
[tex]\[ P(A \cap B) = 0.4 \times 0.8 \][/tex]
Multiplying these numbers together:
[tex]\[ P(A \cap B) = 0.32 \][/tex]
Thus, the probability of both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occurring, [tex]\( P(A \cap B) \)[/tex], is [tex]\( 0.32 \)[/tex].
Therefore, the correct answer is:
C. 0.32
