If A and B are independent events such that `P(A) gt 0, P(B) gt 0`, then (a) `1-P(A//B)` (b) `1-P(A //bar(B))` (c) `(1- P(A uu B))/(P(B))` (d) `(P(A))/(P(B))+(P(bar(A)))/(P(B))=1`

Options

A. `1-P(A//B)`
B. `1-P(A //bar(B))`
C. `(1- P(A uu B))/(P(B))`
D. `(P(A))/(P(B))+(P(bar(A)))/(P(B))=1`

Answer

To solve the problem, we need to analyze the given options based on the properties of independent events A and B.

### Step-by-Step Solution:

1. **Understanding Independent Events**:
- Since A and B are independent events, we have:
\[ P(A \cap B) = P(A) \cdot P(B) \]
- Also, for independent events, the probability of A given B is:
\[ P(A | B) = P(A) \]
- Similarly, the probability of A given not B is:
\[ P(A | \bar{B}) = P(A) \]

2. **Analyzing Option (a)**:
- We need to evaluate \(1 - P(A | B)\):
\[ 1 - P(A | B) = 1 - P(A) \]
- This does not simplify to a known identity involving both A and B.

3. **Analyzing Option (b)**:
- We need to evaluate \(1 - P(A | \bar{B})\):
\[ 1 - P(A | \bar{B}) = 1 - P(A) \]
- This also does not yield a useful identity.

4. **Analyzing Option (c)**:
- We need to evaluate \(\frac{1 - P(A \cup B)}{P(B)}\):
- Using the formula for the union of two independent events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) = P(A) + P(B) - P(A) \cdot P(B) \]
- Thus, we have:
\[ 1 - P(A \cup B) = 1 - (P(A) + P(B) - P(A) \cdot P(B)) = 1 - P(A) - P(B) + P(A) \cdot P(B) \]
- Dividing by \(P(B)\) does not lead to a simplification that matches any of the options.

5. **Analyzing Option (d)**:
- We need to evaluate:
\[ \frac{P(A)}{P(B)} + \frac{P(\bar{A})}{P(B)} = 1 \]
- Since \(P(\bar{A}) = 1 - P(A)\), we can rewrite it as:
\[ \frac{P(A) + (1 - P(A))}{P(B)} = \frac{1}{P(B)} = 1 \]
- This is indeed a true statement.

### Conclusion:
The correct answer is option (d):
\[ \frac{P(A)}{P(B)} + \frac{P(\bar{A})}{P(B)} = 1 \]