In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the…

Consider following events
$A$ : Person chosen is a smoker and non
vegetarian.

$B$ : Person chosen is a smoker and vegetarian.

$C$ : Person chosen is a non-smoker and vegetarian

$E$ : Person chosen has a chest disorder

Given

$P\left(A\right)=\frac{160}{400}P\left(B\right)=\frac{100}{400}P\left(C\right)=\frac{140}{400}$

$P\left(\frac{E}{A}\right)=\frac{35}{100}P\left(\frac{E}{B}\right)=\frac{20}{100}P\left(\frac{E}{C}\right)=\frac{10}{100}$

To find

$P\left(\frac{A}{E}\right)=\frac{P\left(A\right)P\left(\frac{E}{A}\right)}{P\left(A\right)·P\left(\frac{E}{A}\right)+P\left(B\right)·P\left(\frac{E}{B}\right)+P\left(C\right)·P\left(\frac{E}{C}\right)}$

$=\frac{\frac{160}{400}\times \frac{35}{100}}{\frac{160}{400}\times \frac{35}{100}+\frac{100}{400}\times \frac{20}{100}+\frac{140}{400}\times \frac{10}{100}}$

$=\frac{28}{45}$

option $\left(4\right)$