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 :
Answer & explanation
Correct answer: option 1
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 $P(A|X)=\frac{P(A)P(X|A)}{P(A)P(X|A)+P(\overline A)P(X|\overline A)}$
$=\frac{\frac{1}{2}×\frac{3}{5}}{\frac{1}{2}×\frac{3}{5}+\frac{1}{2}×\frac{2}{5}}=\frac{3}{5}$