Question

Two persons, whose probabilities of speaking the truth are $\frac{2}{3}$ and $\frac{5}{6}$ respectively, assert that a specified ticket has been drawn out of a bag containing $15$ tickets. What is the probability of the truth of the assertion?

Answer 1

probability of P1 saying false$=1-\frac{2}{3}=\frac{1}{3}$

probability of P2 saying false$=1-\frac{5}{6}=\frac{1}{6}$

Probability of truth $=\frac{bothP1,P2truth}{(bothP1,P2truth)+(P1false,P2true)+(P1true,P2false)+(bothP1,P2false)}$

$=\frac{\frac{2}{3}\times \frac{5}{6}}{\frac{2}{3}\times \frac{5}{6}+\frac{1}{3}\times \frac{1}{6}+\frac{2}{3}\times \frac{1}{6}+\frac{1}{3}\times \frac{1}{6}}$

$=\frac{5}{7}$