A man A speaks truth in 70% of the cases while another man B speaks truth in 60% of the cases. Find the probability of an event in which they agree with one another.

Explanation

To solve the problem, we need to find the probability that two men, A and B, agree with each other based on their probabilities of speaking the truth. Let's break this down step by step.

### Step 1: Define the Events
Let:
- \( E_1 \): Event that A speaks the truth.
- \( E_2 \): Event that B speaks the truth.

### Step 2: Assign Probabilities
From the problem statement:
- Probability that A speaks the truth:
\[ P(E_1) = 0.7 = \frac{7}{10} \]
- Probability that B speaks the truth:
\[ P(E_2) = 0.6 = \frac{6}{10} \]

### Step 3: Calculate the Probabilities of Speaking False
- Probability that A speaks false (does not speak the truth):
\[ P(E_1') = 1 - P(E_1) = 1 - \frac{7}{10} = \frac{3}{10} \]
- Probability that B speaks false:
\[ P(E_2') = 1 - P(E_2) = 1 - \frac{6}{10} = \frac{4}{10} \]

### Step 4: Identify the Cases for Agreement
A and B can agree in two scenarios:
1. Both A and B speak the truth.
2. Both A and B speak false.

### Step 5: Calculate the Probability of Each Case
1. **Both speak the truth**:
\[ P(E_1 \cap E_2) = P(E_1) \times P(E_2) = \frac{7}{10} \times \frac{6}{10} = \frac{42}{100} \]

2. **Both speak false**:
\[ P(E_1' \cap E_2') = P(E_1') \times P(E_2') = \frac{3}{10} \times \frac{4}{10} = \frac{12}{100} \]

### Step 6: Combine the Probabilities
Since these two cases are mutually exclusive, we can add the probabilities:
\[ P(\text{A and B agree}) = P(E_1 \cap E_2) + P(E_1' \cap E_2') = \frac{42}{100} + \frac{12}{100} = \frac{54}{100} \]

### Step 7: Final Probability
Thus, the probability that A and B agree with one another is:
\[ P(\text{A and B agree}) = 0.54 \]

### Final Answer
The final probability that A and B agree with one another is \( \frac{54}{100} \) or 0.54.

---