The probabilities that A and B speak truth are 4 5 and 3 4 respectively. The probability that they…

Probability of $\mathrm{A}$ speaking truth is $\mathrm{P}\left(\mathrm{A}\right)=\frac{4}{5}$

Probability of $\mathrm{A}$ not speaking truth is $\mathrm{P}\left(\overline{\mathrm{A}}\right)=1-\mathrm{P}\left(\mathrm{A}\right)=1-\frac{4}{5}=\frac{1}{5}$

Probability of $\mathrm{B}$ speaking truth is $\mathrm{P}\left(\mathrm{B}\right)=\frac{3}{4}$

Probability of $\mathrm{B}$ not speaking truth is $\mathrm{P}\left(\overline{\mathrm{B}}\right)=1-\mathrm{P}\left(\mathrm{B}\right)=1-\frac{3}{4}=\frac{1}{4}$

Probability that they contradict each other is$=\mathrm{P}\left(\mathrm{B}\right).\mathrm{P}\left(\overline{\mathrm{A}}\right)+\mathrm{P}\left(\mathrm{A}\right).\mathrm{P}\left(\overline{\mathrm{B}}\right)$

$=\frac{3}{4}\times \frac{1}{5}+\frac{4}{5}\times \frac{1}{4}=\frac{7}{20}$