The odds that X speaks the truth are 3:2 and the odds that Y speaks the truth are 5:3. In what percentage of the cases are they likely to contradict each other?
The probability for X to say the truth is pX = 3/(3+2) = 3/5 and the probability for Y to say the truth is pY = 5/(5+3) = 5/8
The event A that "they contradict each other" means that "X speaks the truth and Y lies or X lies and Y speaks the truth" Hence P(A) = pX*(1-pY) + (1-pX)*pY = 3/5 * 3/8 + 2/5 * 5/8 = 9/40 + 10/40 = 19/40 = 0.475
Thus in 47.5% of cases X and Y contradict each other.