Question

$25 \%$ of the popoulation are smokers. A smoker has $27$ times more chances to develop lung cancer than a non smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac{k}{10}$, Then the value of $k$ is____

Answer 1

$E _1: \text { Smokers } $
$ P \left( E _1\right)=\frac{1}{4} $
$E _2: \text { non-smokers } $
$P \left( E _2\right)=\frac{3}{4}$
E : diagnosed with lung cancer
$ P\left(E / E_1\right)=\frac{27}{28} $
$ P\left(E / E_2\right)=\frac{1}{28}$
$P\left(E_1 / E\right)=\frac{P\left(E_1\right) P\left(E / E_1\right)}{P(E)}$
$ =\frac{\frac{1}{4} \times \frac{27}{28}}{\frac{1}{4} \times \frac{27}{28}+\frac{3}{4} \times \frac{1}{28}}=\frac{\not{27}^9}{\not{30}_{10}}=\frac{9}{10} $
$ K=9$