The probability that A speaks the truth is $\frac{4}{5}$. A coin is tossed and A reports 'Heads'. The probability that it actually showed 'Heads' is :

$\frac{4}{5}$

The correct answer is Option (2) → $\frac{4}{5}$

A → A speaks truth

$P(A)=\frac{4}{5}$,  $P(\bar A)=\frac{1}{5}$

$P(H|A)=\frac{1}{2}$,  $P(H|\bar A)=\frac{1}{2}$

H → Head appears

$P(A|H)=\frac{P(A)P(H|A)}{P(A)P(H|A)+P(\bar A)P(H|\bar A)}$

$=\frac{\frac{4}{5}×\frac{1}{2}}{\frac{4}{5}×\frac{1}{2}+\frac{1}{5}×\frac{1}{2}}=\frac{4}{5}$