This is the application of Bayes' Theorem.
In Bayes' theorem application some event already takes place. Here, the assertion has been made.
Let A denote assertion, T truth and F false. According to the the assertion being true or not true, the specified ticket is drawn or not drawn.
\(P(T/A)=\frac{P(T)P(A/T)}{P(T)P(A/T)+P(F)P(A/F)}\)
While evaluating P(A/T) & P(A/F), both persons need to be considered.
For example, P(A/T)=2/3 x 5/6