Two events A and B are said to be independent if :

Options

A. A and B are mutually exclusive
B. `P(A'B')=[1-p(A)][1-P(B)]`
C. `P(A)=P(B)`
D. `P(A)+P(B)=1`

Answer

To determine the conditions under which two events A and B are independent, we need to analyze the definitions and properties of independent events in probability.

### Step-by-Step Solution:

1. **Understanding Independence of 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:
\[ P(A \cap B) = P(A) \cdot P(B) \]
- This means that the probability of both events occurring together is equal to the product of their individual probabilities.

2. **Mutually Exclusive Events**:
- Events A and B are mutually exclusive if they cannot occur at the same time. In this case:
\[ P(A \cap B) = 0 \]
- For mutually exclusive events, the occurrence of one event means the other cannot occur.

3. **Analyzing the Options**:
- **Option 1**: If A and B are mutually exclusive, then \( P(A \cap B) = 0 \). For independence, we would require \( P(A) \cdot P(B) = 0 \). This can only be true if at least one of the probabilities is zero, which contradicts the idea of independence unless one event is impossible.
- **Conclusion**: This option is incorrect.

4. **Option 2**:
- The expression \( P(A') \cdot P(B') \) refers to the probabilities of the complements of A and B. Using the complement rule:
\[ P(A') = 1 - P(A) \quad \text{and} \quad P(B') = 1 - P(B) \]
- The independence of A and B implies that the complements are also independent:
\[ P(A' \cap B') = P(A') \cdot P(B') \]
- This option can be correct.

5. **Option 3**:
- If \( P(A) = P(B) \), it does not imply independence. Independence is about the relationship of their occurrences, not their probabilities being equal.
- Thus, this option is incorrect.

6. **Final Conclusion**:
- The only correct statement regarding the independence of events A and B is that the complements of A and B are also independent.

### Summary:
- Two events A and B are independent if:
\[ P(A \cap B) = P(A) \cdot P(B) \]
- The only valid option regarding independence from the given choices is the one concerning the complements of A and B.