The probability that A speaks truth is `5/6`, while this probability for B is `4/5`.The probability that they contradict each other when asked to speak on a fact is ___

Answer

To find the probability that A and B contradict each other when asked to speak on a fact, we can follow these steps:

### Step 1: Define the probabilities
Let:
- \( P(A \text{ speaks truth}) = \frac{5}{6} \)
- \( P(A \text{ lies}) = P(A') = 1 - P(A) = 1 - \frac{5}{6} = \frac{1}{6} \)
- \( P(B \text{ speaks truth}) = \frac{4}{5} \)
- \( P(B \text{ lies}) = P(B') = 1 - P(B) = 1 - \frac{4}{5} = \frac{1}{5} \)

### Step 2: Identify the scenarios for contradiction
A and B contradict each other in two scenarios:
1. A speaks the truth and B lies.
2. A lies and B speaks the truth.

### Step 3: Calculate the probability for each scenario
1. **Probability that A speaks truth and B lies:**
\[ P(A \text{ speaks truth} \cap B \text{ lies}) = P(A) \times P(B') = \frac{5}{6} \times \frac{1}{5} = \frac{5}{30} = \frac{1}{6} \]

2. **Probability that A lies and B speaks truth:**
\[ P(A \text{ lies} \cap B \text{ speaks truth}) = P(A') \times P(B) = \frac{1}{6} \times \frac{4}{5} = \frac{4}{30} = \frac{2}{15} \]

### Step 4: Add the probabilities of the two scenarios
To find the total probability that A and B contradict each other, we add the probabilities from the two scenarios:
\[ P(\text{contradict}) = P(A \text{ speaks truth} \cap B \text{ lies}) + P(A \text{ lies} \cap B \text{ speaks truth}) \]
\[ = \frac{1}{6} + \frac{2}{15} \]

### Step 5: Find a common denominator and calculate
The least common multiple of 6 and 15 is 30. We convert both fractions:
\[ \frac{1}{6} = \frac{5}{30}, \quad \frac{2}{15} = \frac{4}{30} \]
Now, adding these gives:
\[ P(\text{contradict}) = \frac{5}{30} + \frac{4}{30} = \frac{9}{30} = \frac{3}{10} \]

### Final Answer
Thus, the probability that A and B contradict each other is \( \frac{3}{10} \).

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