A speaks truth in 75% of the cases while B in 60% of the cases. What is the probability of their contradicting with each other?

Options

A. 0.25
B. 0.55
C. 0.75
D. 0.45

Answer

To find the probability of A and B contradicting each other, we need to consider the scenarios in which one speaks the truth while the other lies.

### Step-by-Step Solution:

1. **Identify the probabilities of truth and lies for A and B:**
- Probability that A speaks the truth (P(A)) = 75% = 0.75
- Probability that A lies (P(A')) = 1 - P(A) = 1 - 0.75 = 0.25
- Probability that B speaks the truth (P(B)) = 60% = 0.60
- Probability that B lies (P(B')) = 1 - P(B) = 1 - 0.60 = 0.40

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

3. **Calculate the probability for each scenario:**
- For Scenario 1 (A is truthful and B is lying):
\[ P(A \text{ and } B') = P(A) \times P(B') = 0.75 \times 0.40 = 0.30 \]
- For Scenario 2 (A is lying and B is truthful):
\[ P(A' \text{ and } B) = P(A') \times P(B) = 0.25 \times 0.60 = 0.15 \]

4. **Add the probabilities of the two scenarios to find the total probability of contradiction:**
\[ P(\text{Contradiction}) = P(A \text{ and } B') + P(A' \text{ and } B) = 0.30 + 0.15 = 0.45 \]

### Final Answer:
The probability of A and B contradicting each other is **0.45** or **45%**.

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