Person A speaks the truth in 75% of the cases and person B speaks the truth in 80% of the cases. If they both independently describe the same incident, what is the probability that their statements will contradict each other?

Introduction / Context:
This question deals with conditional behaviour of two people describing the same incident. A speaks truth with a certain probability and B speaks truth with another probability. Their statements contradict when exactly one of them tells the truth and the other tells a lie. We model this with basic probability of independent events.

Given Data / Assumptions:

• P(A tells truth) = 75% = 0.75, so P(A lies) = 0.25.
• P(B tells truth) = 80% = 0.80, so P(B lies) = 0.20.
• A and B speak independently.
• They contradict each other if one speaks truth and the other lies.

Concept / Approach:
Let T denote telling the truth and L denote lying. Contradiction occurs when A is T and B is L, or A is L and B is T. The total probability of contradiction is the sum of the probabilities of these two mutually exclusive cases. We use multiplication for independent events and then addition because the two cases cannot occur simultaneously.

Step-by-Step Solution:
P(A tells truth) = 0.75, P(A lies) = 0.25. P(B tells truth) = 0.80, P(B lies) = 0.20. Case 1: A tells truth and B lies. P(case 1) = 0.75 * 0.20 = 0.15. Case 2: A lies and B tells truth. P(case 2) = 0.25 * 0.80 = 0.20. P(contradiction) = P(case 1) + P(case 2) = 0.15 + 0.20 = 0.35. 0.35 as a percentage is 35%, that is 35/100.

Verification / Alternative check:
We can also compute the probability that they agree (both tell truth or both lie) and then subtract from 1. P(both truth) = 0.75 * 0.80 = 0.60. P(both lies) = 0.25 * 0.20 = 0.05. So P(agree) = 0.60 + 0.05 = 0.65. Then P(contradiction) = 1 - 0.65 = 0.35, which matches the earlier result.

Why Other Options Are Wrong:
30/100 and 50/100 are different from 35/100 and do not result from the correct multiplication and addition. 45/100 would require a different set of truth and lie probabilities and is not supported by the given data.

Common Pitfalls:
One common mistake is to add probabilities 0.75 and 0.80 directly, ignoring independence and the need to consider separate outcome combinations. Another error is to forget that both truth and both lies lead to agreement, not contradiction. Drawing a simple table of four outcomes (TT, TL, LT, LL) helps keep the logic clear.

Final Answer:
Therefore, the probability that A and B contradict each other is 35/100.