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

Let us define the events for both individuals speaking the truth or lying.

The probability that $$ A $$ speaks the truth is:

$$ P(A) = \frac{4}{5} $$

The probability that $$ A $$ tells a lie is:

$$ P(A') = 1 - P(A) = 1 - \frac{4}{5} = \frac{1}{5} $$

The probability that $$ B $$ speaks the truth is:

$$ P(B) = \frac{3}{4} $$

The probability that $$ B $$ tells a lie is:

$$ P(B') = 1 - P(B) = 1 - \frac{3}{4} = \frac{1}{4} $$

Step 1: Understanding Contradiction

Two people contradict each other if one speaks the truth and the other tells a lie. This can happen in two mutually exclusive ways:

Case 1: $$ A $$ speaks the truth and $$ B $$ lies.

Case 2: $$ A $$ lies and $$ B $$ speaks the truth.

Step 2: Formulating the Probability Equation

Since the actions of $$ A $$ and $$ B $$ are independent events, we calculate the total probability as:

$$ P(\text{Contradiction}) = P(A) \cdot P(B') + P(A') \cdot P(B) $$

Step 3: Substituting the Values

Substitute the calculated probabilities into our formula:

$$ P(\text{Contradiction}) = \left(\frac{4}{5} \cdot \frac{1}{4}\right) + \left(\frac{1}{5} \cdot \frac{3}{4}\right) $$

$$ P(\text{Contradiction}) = \frac{4}{20} + \frac{3}{20} $$

$$ P(\text{Contradiction}) = \frac{7}{20} $$

Therefore, the probability that they contradict each other is $$ \frac{7}{20} $$ (or $$ 35\% $$).