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 :
Answer & explanation
Correct answer: option 1
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
$=\frac{\frac{1}{6} \times \frac{4}{5}}{\frac{1}{6} \times \frac{4}{5}+\frac{5}{6} \times \frac{4}{5}}$
$=\frac{4}{9}$