If a person A speaks the truth in 80% cases and the person B speaks the truth in 75% cases, then the probability that they contradict each other in a statement is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{7}{20}\)
Let A and B also denote the events that A and B speak the truth; they speak independently:
\(\displaystyle P(A) = \frac{4}{5},\quad P(A') = \frac{1}{5}\)
\(\displaystyle P(B) = \frac{3}{4},\quad P(B') = \frac{1}{4}\)
They contradict each other when exactly one of them speaks the truth:
\(P(\text{contradict}) = P(A)P(B') + P(A')P(B)\)
\(\displaystyle = \frac{4}{5}\cdot\frac{1}{4} + \frac{1}{5}\cdot\frac{3}{4}\)
\(\displaystyle = \frac{4}{20} + \frac{3}{20} = \frac{7}{20}\)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{2}{5}\) — Incorrect. \(\frac{2}{5} = \frac{8}{20}\), not \(\frac{7}{20}\).
Option (2) \(\displaystyle\frac{13}{20}\) — Incorrect. This is the probability that they agree (both true or both false): \(\frac{4}{5}\cdot\frac{3}{4} + \frac{1}{5}\cdot\frac{1}{4} = \frac{13}{20}\), the complement of the required event.
Option (3) \(\displaystyle\frac{3}{5}\) — Incorrect. This is the probability that both speak the truth: \(\frac{4}{5}\cdot\frac{3}{4} = \frac{3}{5}\).