A and B are two independent witnesses in a case. The probability that A will speak the truth is x and the probability that B will speak the truth is y. A and B agree on a certain statement. The probability that the statement is true is

Options

A. `(2xy)/(1-x-y+xy)`
B. `(xy)/(1-x-y+2xy)`
C. `(xy)/(1+x+y-xy)`
D. `(2xy)/(1+x+y-xy)`

Answer

To solve the problem, we need to find the probability that the statement agreed upon by two independent witnesses A and B is true, given their respective probabilities of speaking the truth.

### Step-by-Step Solution:

1. **Define Events**:
- Let \( A \) be the event that witness A speaks the truth, with probability \( P(A) = x \).
- Let \( B \) be the event that witness B speaks the truth, with probability \( P(B) = y \).
- Let \( K \) be the event that both witnesses agree on a statement.
- Let \( L \) be the event that both witnesses speak the truth.
- Let \( M \) be the event that both witnesses lie.

2. **Calculate Probabilities**:
- The probability that both A and B speak the truth (event \( L \)) is given by:
\[ P(L) = P(A) \cdot P(B) = x \cdot y \]
- The probability that both A and B lie (event \( M \)) is given by:
\[ P(M) = (1 - P(A)) \cdot (1 - P(B)) = (1 - x)(1 - y) = 1 - x - y + xy \]

3. **Calculate Probability of Agreement**:
- The event \( K \) occurs if both witnesses either speak the truth or both lie. Therefore, the probability of agreement \( P(K) \) can be expressed as:
\[ P(K) = P(L) + P(M) = xy + (1 - x - y + xy) = 1 - x - y + 2xy \]

4. **Use Conditional Probability**:
- We want to find the probability that the statement is true given that both witnesses agree, which is expressed as \( P(L | K) \). According to the formula for conditional probability:
\[ P(L | K) = \frac{P(L \cap K)}{P(K)} \]
- Since \( L \) (both speak the truth) is a subset of \( K \) (both agree), we have \( P(L \cap K) = P(L) = xy \).

5. **Final Calculation**:
- Now substituting the values into the conditional probability formula:
\[ P(L | K) = \frac{P(L)}{P(K)} = \frac{xy}{1 - x - y + 2xy} \]

### Conclusion:
The probability that the statement is true given that both witnesses agree is:
\[ P(L | K) = \frac{xy}{1 - x - y + 2xy} \]