‘A’ speaks the truth in 80% of the cases while ‘B’ in 90% of the cases. The probability that they contradict each other in stating the same statement is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{13}{50}\)
\(\displaystyle P(\text{A true}) = \frac{4}{5},\quad P(\text{A false}) = \frac{1}{5}\)
\(\displaystyle P(\text{B true}) = \frac{9}{10},\quad P(\text{B false}) = \frac{1}{10}\)
They contradict each other when exactly one of them speaks the truth. The two cases are exclusive, and A and B speak independently:
\(\displaystyle P = \frac{4}{5} \times \frac{1}{10} + \frac{1}{5} \times \frac{9}{10}\)
\(\displaystyle = \frac{4}{50} + \frac{9}{50} = \frac{13}{50}\)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{37}{50}\) — Incorrect. This is the probability that they agree (both true or both false): \(\frac{36}{50} + \frac{1}{50} = \frac{37}{50} = 1 - \frac{13}{50}\).
Option (3) \(\displaystyle\frac{18}{25}\) — Incorrect. This is \(\frac{4}{5} \times \frac{9}{10}\), the probability that both speak the truth.
Option (4) \(\displaystyle\frac{9}{25}\) — Incorrect. It equals \(\frac{18}{50}\), not \(\frac{13}{50}\).