Probability that A speaks truth is $\frac{4}{5}$. A coin is tossed. A reports that a head appears. The probability that actually there was head is :

$\frac{4}{5}$

The correct answer is Option (3) → $\frac{4}{5}$

A speaks truth

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

H → Head appears

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

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

$=\frac{4}{5}$