Question

Ten witnesses, each of whom makes but one false statement in six, agree in asserting that a certain event took place, show that the odds are five to one in favour of the truth of their statement, even although the a priori probability of the event is as small as $\frac{1}{5^9+1}$.

Answer 1

Let p be the probability of the event; then the probability that their statement is true is to the probability that it is false as $\frac{{\left(\frac{5}{6}\right)}^{10}p}{(1-p){\left(\frac{1}{6}\right)}^{10}}$

$\therefore \frac{5^{10}p}{1-p}$ represents the odds in favor of the event. Now in order that the odds in facor of the event may be atleast five to one, we must have $\frac{5^{10}p}{1-p}\ge 5$;

$⟹5^{10}p\ge 5-5p⟹5^{9}p\ge 1-p⟹5^{9}p+p\ge 1⟹p(5^9+1)\ge 1⟹p\ge \frac{1}{5^9+1}$