Let `A, B, C` be three events such that A and B are independent and `P(C) = 0`, then events `A, B ,C` are

Options

A. A and C are independent
B. B and C are independent
C. A, B and C are independent
D. All of these

Answer

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

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- We know that events A and B are independent.
- We also know that the probability of event C, denoted as P(C), is equal to 0.

2. **Implications of P(C) = 0:**
- Since P(C) = 0, this means that there are no favorable outcomes for event C. In other words, event C cannot occur.

3. **Independence of Events:**
- Two events are considered independent if the occurrence of one does not affect the occurrence of the other.
- Since event C has no outcomes (P(C) = 0), it cannot affect the occurrence of events A or B.

4. **Conclusion about Independence:**
- Because C has no outcomes, we can conclude that:
- A and C are independent.
- B and C are independent.
- This is because the occurrence of A or B does not influence the occurrence of C, which cannot happen.

5. **Final Conclusion:**
- Since A and B are independent, and we have established that A and C, as well as B and C, are also independent, we can conclude that all three events A, B, and C are independent of each other.

6. **Answer:**
- Therefore, the answer is that events A, B, and C are independent.