A speaks truth in 75% cases and B in 80% of the cases. In what percentage of cases are they likely to contradict each other, in narrating the same incident?

### Concept & Logic Two people contradict each other when one speaks the truth and the other lies. We must calculate the probability of two mutually exclusive scenarios and add them together. $$ P(\text{Contradiction}) = P(A_{\text{truth}})P(B_{\text{lie}}) + P(A_{\text{lie}})P(B_{\text{truth}}) $$ ### Step-by-Step Solution * Let $P(A)$ be the probability A speaks truth: $P(A) = 75\% = \frac{75}{100} = \frac{3}{4}$ * Probability A lies: $P(\bar{A}) = 1 - \frac{3}{4} = \frac{1}{4}$ * Let $P(B)$ be the probability B speaks truth: $P(B) = 80\% = \frac{80}{100} = \frac{4}{5}$ * Probability B lies: $P(\bar{B}) = 1 - \frac{4}{5} = \frac{1}{5}$ * They contradict if (A speaks truth AND B lies) OR (A lies AND B speaks truth). * Required Probability $= [P(A) \times P(\bar{B})] + [P(\bar{A}) \times P(B)]$ * $= \left(\frac{3}{4} \times \frac{1}{5}\right) + \left(\frac{1}{4} \times \frac{4}{5}\right)$ * $= \frac{3}{20} + \frac{4}{20} = \frac{7}{20}$ * Convert back to percentage: $\frac{7}{20} \times 100\% = 7 \times 5\% = 35\%$ ### Exam Strategy & Shortcut Keep values in decimals if you prefer avoiding fractions. $P(A) = 0.75$, $P(\bar{A}) = 0.25$, $P(B) = 0.8$, $P(\bar{B}) = 0.2$. Contradiction $= (0.75 \times 0.2) + (0.25 \times 0.8) = 0.15 + 0.20 = 0.35 = 35\%$. ### Common Pitfall Misunderstanding "contradict". Some students calculate the probability they both lie, thinking a lie is a contradiction. Contradiction strictly means opposing statements (Truth vs. Lie). ### Final Answer Therefore, the correct answer is **35%**.