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

Answer: 9

$E_1:\text{ Smokers }$
$P\left(E_1\right)=\frac{1}{4}$
$E_2:\text{ non-smokers }$
$P\left(E_2\right)=\frac{3}{4}$
E : diagnosed with lung cancer
$P\left(E/E_1\right)=\frac{27}{28}$
$P\left(E/E_2\right)=\frac{1}{28}$
$P\left(E_1/E\right)=\frac{P\left(E_1\right)P\left(E/E_1\right)}{P(E)}$
$=\frac{\frac{1}{4}\times \frac{27}{28}}{\frac{1}{4}\times \frac{27}{28}+\frac{3}{4}\times \frac{1}{28}}=\frac{/{27}^9}{/{30}_{10}}=\frac{9}{10}$
$K=9$