If A and B are two independent events and P( C) = 0, then A, B, C are :

Options

A. independent
B. dependent
C. not pairwise independent
D. None of these

Answer

To solve the question, we need to analyze the relationship between the events A, B, and C given that A and B are independent events and the probability of event C is 0.

### Step-by-Step Solution:

1. **Understanding Independent Events**:
- Two events A and B are said to be independent if the occurrence of one does not affect the occurrence of the other. Mathematically, this is expressed as:
\[ P(A \cap B) = P(A) \cdot P(B) \]

2. **Probability of Event C**:
- We are given that \( P(C) = 0 \). This means that event C cannot occur. In probability theory, an event with a probability of 0 is often referred to as an impossible event.

3. **Analyzing the Relationship**:
- Since A and B are independent, we need to determine the relationship of event C with A and B.
- The independence of A and B does not directly influence the probability of C. However, since C has a probability of 0, it does not affect the independence of A and B.

4. **Independence of C with A and B**:
- For C to be independent of A and B, we need to check if:
\[ P(A \cap C) = P(A) \cdot P(C) \quad \text{and} \quad P(B \cap C) = P(B) \cdot P(C) \]
- Since \( P(C) = 0 \), the right-hand side of both equations becomes 0:
\[ P(A \cap C) = P(A) \cdot 0 = 0 \quad \text{and} \quad P(B \cap C) = P(B) \cdot 0 = 0 \]
- Therefore, \( P(A \cap C) = 0 \) and \( P(B \cap C) = 0 \) hold true.

5. **Conclusion**:
- Since A and B are independent events and C has a probability of 0, we conclude that A, B, and C are independent events.

### Final Answer:
Thus, the correct answer is that A, B, and C are independent events.

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