If A and B are independent events such that `P(A) gt0, P(B) gt 0`, then

Options

A. A and B are mutually exclusive
B. A and `overline(B)` are dependent
C. `overline(A)` and B are dependent
D. `P(A//B)+P(overline(A)//B)=1`.

Answer

To solve the problem, we need to analyze the relationship between the probabilities of independent events A and B.

### Step-by-Step Solution:

1. **Understanding Independence**:
Since A and B are independent events, we know that:
\[ P(A \cap B) = P(A) \cdot P(B) \]

2. **Complement Events**:
The complements of A and B, denoted as \( A' \) and \( B' \), are also independent. Thus:
\[ P(A' \cap B) = P(A') \cdot P(B) \]

3. **Conditional Probability**:
We can express the conditional probabilities:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \quad \text{and} \quad P(A' | B) = \frac{P(A' \cap B)}{P(B)} \]

4. **Adding the Conditional Probabilities**:
We want to find:
\[ P(A | B) + P(A' | B) \]
Substituting the expressions from step 3:
\[ P(A | B) + P(A' | B) = \frac{P(A \cap B)}{P(B)} + \frac{P(A' \cap B)}{P(B)} \]

5. **Combining the Fractions**:
Since the denominators are the same, we can combine the fractions:
\[ P(A | B) + P(A' | B) = \frac{P(A \cap B) + P(A' \cap B)}{P(B)} \]

6. **Using the Law of Total Probability**:
The events \( A \) and \( A' \) cover all possibilities in relation to \( B \):
\[ P(A \cap B) + P(A' \cap B) = P(B) \]
Therefore, we can substitute this into our equation:
\[ P(A | B) + P(A' | B) = \frac{P(B)}{P(B)} = 1 \]

7. **Conclusion**:
Thus, we conclude that:
\[ P(A | B) + P(A' | B) = 1 \]
The correct option is D.