A speaks truth in 75% of cases and B in 80% of cases. In what percentage of…

Concept: When two people independently narrate the same incident, they contradict each other only when exactly one of them speaks the truth and the other lies -- never when both are truthful or both are lying (that is agreement, not contradiction). For truth-probabilities P(A) and P(B), P(contradiction) = P(A) × P(B') + P(A') × P(B), the sum of the two mutually exclusive ways to get exactly one truth-teller (A' and B' denote A and B lying, respectively).

Here P(A) = 75% = 3/4, so P(A lies) = 1/4. P(B) = 80% = 4/5, so P(B lies) = 1/5.

A and B contradict each other in exactly two mutually exclusive ways:

1. A tells the truth and B lies: P(A) × P(B') = 3/4 × 1/5 = 3/20.

2. A lies and B tells the truth: P(A') × P(B) = 1/4 × 4/5 = 4/20.

Adding the two mutually exclusive cases: P(contradiction) = 3/20 + 4/20 = 7/20 = 35%.

Cross-check: the complementary event is that A and B agree -- both truthful or both lying -- P(agree) = P(A) × P(B) + P(A') × P(B') = (3/4 × 4/5) + (1/4 × 1/5) = 12/20 + 1/20 = 13/20 = 65%. Since contradiction and agreement are complementary, 35% + 65% = 100%, confirming the result.