There are two persons A and B such that the chances of B speaking truth of A and A speaks truth in more than 25% cases.
Statement-1: If A and B contradict each other in narrating the same statement with probability `1//2`, then it is certain that B never tells a lie.
Statement-2: The probability that A speaks truth is `1//2`.

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Explanation

To solve the problem, we will analyze the given statements and the relationships between the probabilities of A and B speaking the truth.

### Step 1: Understanding the Statements
We have two statements:
- **Statement 1**: If A and B contradict each other in narrating the same statement with probability 1/2, then it is certain that B never tells a lie.
- **Statement 2**: The probability that A speaks the truth is 1/2.

### Step 2: Setting Up the Probabilities
Let:
- \( P(A) \) = Probability that A speaks the truth.
- \( P(B) \) = Probability that B speaks the truth.

From the problem, we know:
- \( P(A) > 0.25 \)
- \( P(A) = 0.5 \) (as per Statement 2).

### Step 3: Analyzing Contradictions
When A and B contradict each other, it can happen in two ways:
1. A speaks the truth and B lies.
2. A lies and B speaks the truth.

The probability of contradiction can be expressed as:
\[ P(\text{Contradiction}) = P(A \text{ is true}) \cdot P(B \text{ is false}) + P(A \text{ is false}) \cdot P(B \text{ is true}) \]

This can be mathematically represented as:
\[ P(\text{Contradiction}) = P(A) \cdot (1 - P(B)) + (1 - P(A)) \cdot P(B) \]

### Step 4: Substituting Values
Substituting \( P(A) = 0.5 \):
\[ P(\text{Contradiction}) = 0.5 \cdot (1 - P(B)) + (1 - 0.5) \cdot P(B) \]
\[ = 0.5 \cdot (1 - P(B)) + 0.5 \cdot P(B) \]
\[ = 0.5 \]

### Step 5: Conclusion for Statement 1
Since the contradiction probability equals 1/2, we need to check if this implies that \( P(B) = 1 \) (i.e., B never tells a lie).

If \( P(B) = 1 \):
- Then \( P(B \text{ is false}) = 0 \).
- The contradiction would only occur when A lies, which is \( 0.5 \cdot 0 + 0.5 \cdot 1 = 0.5 \), confirming that B never lies.

Thus, **Statement 1 is true**.

### Step 6: Conclusion for Statement 2
Since \( P(A) = 0.5 \) is consistent with the condition that \( P(A) > 0.25 \), **Statement 2 is also true**.

### Final Conclusion
Both statements are true, and Statement 2 correctly explains Statement 1.