The probability that A speaks truth is $\frac{4}{5}$ , while this probability for B is $\frac{3}{4}$ . The probability that they contradict each other when asked to speak on a fact is

$\frac{7}{20}$

Let E be the event that B speaks truth and F be the event that A speaks truth.

Now $P(E)=\frac{75}{100}=\frac{3}{4}$ and $P(F)=\frac{80}{100}=\frac{4}{5}.$

∴ P (A and B contradict each other)

$= P[E ∩ \overline{F}) ∪ (\overline{E} ∩ F)]= P(E). P(\overline{F})+ P(\overline{E}). P(F) = \frac{3}{4}×\frac{1}{5}+\frac{1}{4}×\frac{4}{5}=\frac{7}{20}.$