A speaks truth in `75%` of cases and B in `80%` of cases. The percentage of cases theyare likely to contradict each other in stating the same fact, is

Options

A. `30%`
B. `35%`
C. `45%`
D. `25%`

Answer

To solve the problem step by step, we will calculate the probabilities of A and B speaking the truth and the falsehood, and then find the probability of them contradicting each other.

### Step 1: Determine the probabilities of A and B speaking truth and falsehood.

- A speaks the truth in 75% of cases. Therefore, the probability of A speaking the truth (P(A_T)) is:
\[ P(A_T) = \frac{75}{100} = \frac{3}{4} \]

- The probability of A speaking false (P(A_F)) is:
\[ P(A_F) = 1 - P(A_T) = 1 - \frac{3}{4} = \frac{1}{4} \]

- B speaks the truth in 80% of cases. Therefore, the probability of B speaking the truth (P(B_T)) is:
\[ P(B_T) = \frac{80}{100} = \frac{4}{5} \]

- The probability of B speaking false (P(B_F)) is:
\[ P(B_F) = 1 - P(B_T) = 1 - \frac{4}{5} = \frac{1}{5} \]

### Step 2: Determine the probability of A and B contradicting each other.

A and B will contradict each other in two scenarios:
1. A speaks the truth and B speaks false.
2. A speaks false and B speaks the truth.

We can denote the event of contradiction as E. Thus, we can express the probability of E (P(E)) as:
\[ P(E) = P(A_T) \cdot P(B_F) + P(A_F) \cdot P(B_T) \]

Substituting the values we calculated:
\[ P(E) = P(A_T) \cdot P(B_F) + P(A_F) \cdot P(B_T) = \left(\frac{3}{4} \cdot \frac{1}{5}\right) + \left(\frac{1}{4} \cdot \frac{4}{5}\right) \]

### Step 3: Calculate the individual probabilities.

Calculating the first term:
\[ \frac{3}{4} \cdot \frac{1}{5} = \frac{3}{20} \]

Calculating the second term:
\[ \frac{1}{4} \cdot \frac{4}{5} = \frac{4}{20} \]

### Step 4: Combine the probabilities.

Now, we can combine these two probabilities:
\[ P(E) = \frac{3}{20} + \frac{4}{20} = \frac{7}{20} \]

### Step 5: Convert the probability into percentage.

To find the percentage of cases in which A and B contradict each other, we multiply the probability by 100:
\[ \text{Percentage} = P(E) \times 100 = \frac{7}{20} \times 100 = 35\% \]

### Conclusion

The percentage of cases in which A and B are likely to contradict each other is **35%**.