25% of the population are smokers.  A smokers 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
k/10 . Then the value of k is …………….. 

Probability of a person being smoker $\mathrm{P}\left(\mathrm{S}\right)=25/100=1/4$
Probability of a person being non smokes $\mathrm{P}\left(\overline{\mathrm{S}}\right)=1-1/4=3/4$
Probability of anon smoker diagnosed with cancer $\mathrm{P}(\mathrm{C}/\overline{\mathrm{S}})=\mathrm{P}$
Probability of a smoker diagnosed with cancer $\mathrm{P}(\mathrm{C}/\mathrm{S})=27\mathrm{P}$

$\mathrm{P}\left(\frac{\text{ Person is smo ker }}{\text{ P erson diagnosed with cancer }}\right)=P\left(\frac{\mathrm{S}}{\mathrm{C}}\right)=\frac{\mathrm{P}(\mathrm{S}\cap \mathrm{C})}{\mathrm{P}\left(\mathrm{C}\right)}$

$\frac{\mathrm{P}\left(\mathrm{S}\right)\cdot \mathrm{P}\left(\frac{\mathrm{C}}{\mathrm{S}}\right)}{\mathrm{P}\left(\mathrm{S}\right)\cdot \mathrm{P}\left(\frac{\mathrm{C}}{\mathrm{S}}\right)+\mathrm{P}\left(\overline{\mathrm{S}}\right)\cdot \mathrm{P}\left(\frac{\mathrm{C}}{\overline{\mathrm{S}}}\right)}=\frac{\frac{1}{4}\cdot 27\mathrm{P}}{\frac{1}{4}27\mathrm{P}+\frac{3}{4}\mathrm{P}}=\frac{\frac{\mathrm{P}}{4}\cdot 27}{\frac{\mathrm{P}}{4}(27+3)}=\frac{27}{30}=\frac{9}{10}$