A' speaks truth in `60%` cases and 'B' in `70%` cases. In what percentages of cases are they likely to contradict each other in stating the same fact?

Answer

To solve the problem of finding the percentage of cases where A and B contradict each other in stating the same fact, we can follow these steps:

### Step 1: Define the probabilities
Let:
- \( P(A) \) be the probability that A speaks the truth. Given \( P(A) = 60\% = \frac{60}{100} = 0.6 \).
- \( P(B) \) be the probability that B speaks the truth. Given \( P(B) = 70\% = \frac{70}{100} = 0.7 \).

### Step 2: Calculate the probabilities of not speaking the truth
- The probability that A does not speak the truth, denoted as \( P(A') \):
\[ P(A') = 1 - P(A) = 1 - 0.6 = 0.4 \]

- The probability that B does not speak the truth, denoted as \( P(B') \):
\[ P(B') = 1 - P(B) = 1 - 0.7 = 0.3 \]

### Step 3: Identify the scenarios for contradiction
A and B can contradict each other in two scenarios:
1. A speaks the truth and B does not.
2. A does not speak the truth and B does.

### Step 4: Calculate the probabilities for each scenario
1. **A speaks the truth and B does not**:
\[ P(A \text{ speaks truth}) \times P(B' \text{ does not speak truth}) = P(A) \times P(B') = 0.6 \times 0.3 = 0.18 \]

2. **A does not speak the truth and B does**:
\[ P(A' \text{ does not speak truth}) \times P(B \text{ speaks truth}) = P(A') \times P(B) = 0.4 \times 0.7 = 0.28 \]

### Step 5: Calculate the total probability of contradiction
Now, we add the probabilities from both scenarios:
\[ P(\text{Contradiction}) = P(A \text{ speaks truth}) \times P(B') + P(A') \times P(B) = 0.18 + 0.28 = 0.46 \]

### Step 6: Convert to percentage
To express this probability as a percentage:
\[ P(\text{Contradiction in percentage}) = 0.46 \times 100\% = 46\% \]

### Final Answer
The percentage of cases in which A and B are likely to contradict each other is **46%**.

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