Let A,B, C be three mutually independent events. Consider the two statements `S_(1)and S_(2).`
`{:(S_(1):A and B nnC "are independent.",),(S_(2):A and B nnC "are independent.",):}` Then

Options

A. both `S_(1) and S_(2)` are true
B. only `S_(1)` is true
C. only `S_(2)` is true
D. neither `S_(1)` nor `S_(2)` is true

Explanation

To solve the problem, we need to analyze the independence of the events A, B, and C, and the statements S1 and S2 regarding their intersections.

### Step-by-Step Solution:

1. **Understanding Independence**:
Since A, B, and C are mutually independent events, we have:
- \( P(A \cap B) = P(A) \cdot P(B) \)
- \( P(B \cap C) = P(B) \cdot P(C) \)
- \( P(C \cap A) = P(C) \cdot P(A) \)

2. **Analyzing Statement S1**:
We need to check if A and \( B \cap C \) are independent. For A and \( B \cap C \) to be independent, we need to show:
\[ P(A \cap (B \cap C)) = P(A) \cdot P(B \cap C) \]
Using the property of independence:
\[ P(A \cap (B \cap C)) = P(A) \cdot P(B \cap C) = P(A) \cdot (P(B) \cdot P(C)) \]
Therefore, we can conclude that:
\[ P(A \cap (B \cap C)) = P(A) \cdot P(B) \cdot P(C) \]
This shows that A and \( B \cap C \) are independent.

3. **Analyzing Statement S2**:
Now, we need to check if A and \( B \cap C \) are independent. For A and \( B \cap C \) to be independent, we need to show:
\[ P(A \cap (B \cap C)) = P(A) \cdot P(B \cap C) \]
We already established that:
\[ P(A \cap (B \cap C)) = P(A) \cdot P(B) \cdot P(C) \]
And since \( P(B \cap C) = P(B) \cdot P(C) \), we can substitute:
\[ P(A \cap (B \cap C)) = P(A) \cdot (P(B) \cdot P(C)) \]
Thus, A and \( B \cap C \) are also independent.

4. **Conclusion**:
Both statements S1 and S2 are true. Therefore, we conclude that:
- S1: A and \( B \cap C \) are independent.
- S2: A and \( B \cap C \) are independent.

### Final Answer:
Both statements S1 and S2 are true.

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