To find the probability of both independent events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occurring, denoted as [tex]\( P(A \cap B) \)[/tex], we use the rule for the probability of the intersection of two independent events. This rule states that for two independent events [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ P(A \cap B) = P(A) \times P(B) \][/tex]
Given the probabilities:
[tex]\[ P(A) = 0.7 \][/tex]
[tex]\[ P(B) = 0.8 \][/tex]
We multiply these probabilities together:
[tex]\[ P(A \cap B) = 0.7 \times 0.8 \][/tex]
Carrying out the multiplication:
[tex]\[ 0.7 \times 0.8 = 0.56 \][/tex]
Therefore, the probability that both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occur is:
[tex]\[ P(A \cap B) = 0.56 \][/tex]
So, the correct answer is:
D. 0.56
