Let the event that $A$ speaks truth be $T_A$ and that $B$ speaks truth be $T_B$.

Given: $P(T_A)=\frac{4}{5}, \qquad P(T_B)=\frac{3}{4}$

So, probabilities of lying are: $P(L_A)=1-\frac{4}{5}=\frac{1}{5}, \qquad P(L_B)=1-\frac{3}{4}=\frac{1}{4}$

They will contradict each other on a fact only in the following two cases:

• $A$ tells truth and $B$ lies
• $A$ lies and $B$ tells truth

Assuming their statements are independent, the required probability is: $P(T_A \cap L_B) + P(L_A \cap T_B)$

Compute each term: $P(T_A \cap L_B)=\frac{4}{5}\cdot\frac{1}{4}=\frac{1}{5}$ $P(L_A \cap T_B)=\frac{1}{5}\cdot\frac{3}{4}=\frac{3}{20}$

Add them: $\frac{1}{5}+\frac{3}{20}=\frac{4}{20}+\frac{3}{20}=\frac{7}{20}$

Therefore, the probability that they contradict each other is: $\boxed{\frac{7}{20}}$

Option check:

• A: $\frac{4}{5}$ ✗
• B: $\frac{1}{5}$ ✗
• C: $\frac{7}{20}$ ✓
• D: $\frac{3}{20}$ ✗