Certainly! Let's go through the problem step by step.
1. Understanding Event Complements:
- We are given [tex]\(P(A) = 0.4\)[/tex].
- The complement of event [tex]\(A\)[/tex], denoted as [tex]\(A^C\)[/tex], is the event that [tex]\(A\)[/tex] does not occur.
- The probability of the complement of [tex]\(A\)[/tex] is given by:
[tex]\[
P(A^C) = 1 - P(A)
\][/tex]
2. Calculate the Probability of the Complement of [tex]\(A\)[/tex]:
[tex]\[
P(A^C) = 1 - 0.4 = 0.6
\][/tex]
3. Independence of Events:
- Given events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent.
- For independent events, the probability of the intersection of [tex]\(A^C\)[/tex] and [tex]\(B\)[/tex] is the product of their individual probabilities:
[tex]\[
P(A^C \cap B) = P(A^C) \times P(B)
\][/tex]
4. Calculate the Intersection Probability:
[tex]\[
P(A^C \cap B) = 0.6 \times 0.1 = 0.06
\][/tex]
Thus, the probability [tex]\(P(A^C \cap B)\)[/tex] is [tex]\(0.06\)[/tex].
Therefore, the correct answer is:
C. 0.06
