Official solution
The probability of speaking truth by A, P(A) = \[{4 \over 5}\]. The probability of not speaking truth by A, P(\[\overline A \]) = 1 - \[{4 \over 5} = {1 \over 5}\].
The probability of speaking truth by B, P(B) = \[{3 \over 4}\]. The probability of not speaking truth of B, P(\[\overline B \]) \[ = {1 \over 4}\].
The probability that they contradict each other
\[ = P(A) \times P(\overline B ) + P(\overline A ) \times P(B)\]
\[ = {4 \over 5} \times {1 \over 4} + {1 \over 5} \times {3 \over 4}\]
\[ = {1 \over 5} + {3 \over {20}} = {7 \over {20}}\]