Probability that A speaks truth is $\frac{3}{5}$. A coins is tosses. A reports that a head appears. The probability that actually there was a head is :

$\frac{3}{5}$

The correct answer is option (1) → $\frac{3}{5}$

A → A speaks truth

$P(A)=\frac{3}{5}⇒P(\overline A)=\frac{2}{5}$

X → head comes up

$P(X|A)=\frac{1}{2}$, $P(X|\overline A)=\frac{2}{5}$

so $P(A|X)=\frac{P(A)P(X|A)}{P(A)P(X|A)+P(\overline A)P(X|\overline A)}$

$=\frac{\frac{1}{2}×\frac{3}{5}}{\frac{1}{2}×\frac{3}{5}+\frac{1}{2}×\frac{2}{5}}=\frac{3}{5}$