Aman A speaks truth in 80% of the cases and another man B in 90% of the cases. While stating the same fact, what is the probability that they contradict?

Options

A. `(37)/(50)`
B. `(13)/(50)`
C. `(16)/(50)`
D. None of these

Answer

To find the probability that Aman A and man B contradict each other while stating the same fact, we can break down the problem into steps.

### Step-by-Step Solution:

1. **Identify the probabilities of speaking the truth:**
- Probability that A speaks the truth (P(A)) = 80% = 0.8
- Probability that B speaks the truth (P(B)) = 90% = 0.9

2. **Calculate the probabilities of lying:**
- Probability that A lies (P(A')) = 1 - P(A) = 1 - 0.8 = 0.2
- Probability that B lies (P(B')) = 1 - P(B) = 1 - 0.9 = 0.1

3. **Determine the scenarios for contradiction:**
- Scenario 1: A speaks the truth and B lies.
- Scenario 2: A lies and B speaks the truth.

4. **Calculate the probability for each scenario:**
- **Scenario 1:** Probability that A speaks the truth and B lies:
\[ P(A \text{ and } B') = P(A) \times P(B') = 0.8 \times 0.1 = 0.08 \]
- **Scenario 2:** Probability that A lies and B speaks the truth:
\[ P(A' \text{ and } B) = P(A') \times P(B) = 0.2 \times 0.9 = 0.18 \]

5. **Add the probabilities of the two scenarios to find the total probability of contradiction:**
\[ P(\text{Contradiction}) = P(A \text{ and } B') + P(A' \text{ and } B) = 0.08 + 0.18 = 0.26 \]

6. **Convert the probability into a fraction:**
- 0.26 can be expressed as \( \frac{26}{100} \), which simplifies to \( \frac{13}{50} \).

### Final Answer:
The probability that Aman A and man B contradict each other is \( \frac{13}{50} \).

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