The probability that A speaks truth is 4 / 5 while this probability for B is 3 / 4. The probability that they contradict each other when asked to speak on a fact is
\[ \begin{aligned} &\text{The probability of speaking truth of A is } P(A) = \frac{4}{5} \\ &\text{The probability of not speaking truth of A is } P(A') = 1 - \frac{4}{5} = \frac{1}{5} \\ &\text{The probability of speaking truth of B is } P(B) = \frac{3}{4} \\ &\text{The probability of not speaking truth of B is } P(B') = 1 - \frac{3}{4} = \frac{1}{4} \\ &\text{The probability that they contradict each other} \\ &= P(A) \cdot P(B') + P(A') \cdot P(B) \\ &= \frac{4}{5} \times \frac{1}{4} + \frac{1}{5} \times \frac{3}{4} \\ &= \frac{4}{20} + \frac{3}{20} \\ &= \frac{7}{20} \end{aligned} \]
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