To determine whether two events, A and B, are independent, we must check if the product of their individual probabilities [tex]\(P(A)\)[/tex] and [tex]\(P(B)\)[/tex] is equal to the probability of both events occurring together, which is [tex]\(P(A \cap B)\)[/tex].
Given:
- [tex]\(P(A) = 0.4\)[/tex]
- [tex]\(P(B) = 0.2\)[/tex]
- [tex]\(P(A \cap B) = 0.1\)[/tex]
Step-by-step solution:
1. Calculate the product of [tex]\(P(A)\)[/tex] and [tex]\(P(B)\)[/tex]:
[tex]\[
P(A) \times P(B) = 0.4 \times 0.2 = 0.08
\][/tex]
2. Compare the product [tex]\(P(A) \times P(B)\)[/tex] to [tex]\(P(A \cap B)\)[/tex]:
[tex]\[
P(A \times B) = 0.08
\][/tex]
[tex]\[
P(A \cap B) = 0.1
\][/tex]
Since [tex]\(P(A) \times P(B)\)[/tex] equals 0.08 and [tex]\(P(A \cap B)\)[/tex] equals 0.1, we notice that:
[tex]\[
P(A) \times P(B) \neq P(A \cap B)
\][/tex]
Therefore, the reason the events A and B are not independent is that the product of [tex]\(P(A)\)[/tex] and [tex]\(P(B)\)[/tex] is not equal to [tex]\(P(A \cap B)\)[/tex].
Hence, the correct answer is:
- The product of [tex]\(P(A)\)[/tex] and [tex]\(P(B)\)[/tex] is not equal to [tex]\(P(A \cap B)\)[/tex].
