Concept:

Let A₁, A₂, ..., A_n be n mutually exclusive and exhaustive events of the sample space S, and A is an event that can occur with any of these events. Then:

• \[ P(A_i | A) = \frac{P(A_i) P(A | A_i)}{\sum_{i=1}^{n} P(A_i) P(A | A_i)} \]

Calculations:

Consider K as the event that both A and B agree.

Let T be the event that both A and B speak the truth:

\[ P(T) = xy \]

Let L be the event that both A and B lie:

\[ P(L) = (1 - x)(1 - y) \]

To find: The probability that the statement is true:

\[ P(T | K) \]

Since K is the event that both of them agree, we use Bayes' Theorem:

\[ P(T | K) = \frac{P(T) P(K | T)}{P(T) P(K | T) + P(L) P(K | L)} \]

\[ P(T | K) = \frac{xy}{xy + (1 - x)(1 - y)} \]