To test whether events [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are independent, we need to compare [tex]\( P(B \cap C) \)[/tex] with [tex]\( P(B) \cdot P(C) \)[/tex].
### Step-by-Step Solution:
1. Identify the probabilities from the table:
- [tex]\( P(B) = 0.30 \)[/tex]
- [tex]\( P(C) = 0.30 \)[/tex]
2. Calculate the joint probability [tex]\( P(B \cap C) \)[/tex]:
Since [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] are mutually exclusive, there is no overlap between these events. Hence,
[tex]\[
P(B \cap C) = 0.0
\][/tex]
3. Calculate the product [tex]\( P(B) \cdot P(C) \)[/tex]:
[tex]\[
P(B) \cdot P(C) = 0.30 \times 0.30 = 0.09
\][/tex]
4. Compare [tex]\( P(B \cap C) \)[/tex] with [tex]\( P(B) \cdot P(C) \)[/tex]:
[tex]\[
P(B \cap C) = 0.0 \quad \text{and} \quad P(B) \cdot P(C) = 0.09
\][/tex]
Since [tex]\( P(B \cap C) \neq P(B) \cdot P(C) \)[/tex], the events [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are not independent.
### Answer:
A. No, they are not independent because [tex]\( P(B \cap C) \neq P(B) \cdot P(C) \)[/tex]. [tex]\( P(B \cap C) = 0.0 \)[/tex] and [tex]\( P(B) \cdot P(C) = 0.09 \)[/tex].
So, the filled in answer choice is:
```markdown
A. No, they are not independent because [tex]\( P(B \cap C) \neq P(B) \cdot P(C) \)[/tex]. [tex]\( P(B \cap C) = 0.0 \)[/tex] and [tex]\( P(B) \cdot P(C) = 0.09 \)[/tex].
```
