The probability that A speaks the truth is $\frac{4}{5}$. He throws a die and reports that it is a five. The probability that it is actually a five is :

$\frac{4}{9}$

A → a speaks truth

$\bar{A}$ → A lies

P(A) = $\frac{4}{5}$

$P(\bar{A}) = \frac{1}{5}$

E → 5 has come

(A → reports it as 5)

$\bar{A}$ → doesn't report it as 5

$\bar{E}$ → doesn't come

so P(E/A)

by bayes theorm

P(E/A) = $\frac{P(E) × P(A/E)}{P(E) × P(A/E) + P(\bar{E})P(A/\bar{E})}$

A/E → a speaks truth that 5 has appeared

$A/\bar{E}$ → A lies that 5 has appeared

$=\frac{\frac{1}{6} \times \frac{4}{5}}{\frac{1}{6} \times \frac{4}{5}+\frac{5}{6} \times \frac{4}{5}}$

$=\frac{4}{9}$