A person X speaks the truth 4 out of 5 times and person Y speaks...

Let $P(X)$ be the probability that person $X$ speaks the truth and $P(Y)$ be the probability that person $Y$ speaks the truth.

$P(X) = \dfrac{4}{5}$

$P(Y) = \dfrac{5}{6}$

The probability that $X$ lies is $P(X') = 1 - \dfrac{4}{5} = \dfrac{1}{5}$.

The probability that $Y$ lies is $P(Y') = 1 - \dfrac{5}{6} = \dfrac{1}{6}$.

They contradict each other when one speaks the truth and the other lies. The probability of this event is given by:

$P(\text{contradiction}) = P(X)P(Y') + P(X')P(Y)$

$P(\text{contradiction}) =$\left(\dfrac{4}{5}\right)\left(\dfrac{1}{6}\right) + \left(\dfrac{1}{5}\right)\left(\dfrac{5}{6}\right)

$P($\text{contradiction}) = \dfrac{4}{30} + \dfrac{5}{30} = \dfrac{9}{30} = \dfrac{3}{10}

Answer: 3/10