Question

$10$ witnesses each of whose probability of speaking the truth is $\frac{5}{6}$. Asserts that the certain event took place, if the probability of this event before their statement is $\frac{1}{1+5^9}$, find the probability of the truth of their assertion.

Answer 1

Probability that all 10 would say the truth = ${\left(\frac{5}{6}\right)}^{10}$

Probability that all 10 would lie = ${\left(\frac{1}{6}\right)}^{10}$

Hence, probability that the event occurred $=\frac{(\frac{5}{6}{)}^{10}}{(\frac{5}{6}{)}^{10}+(\frac{1}{6}{)}^{10}}=\frac{5^{10}}{1+5^{10}}$