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

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

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

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

To find

$P\left(\dfrac{A}{E}\right) = \dfrac{P\left(A\right)P\left(\dfrac{E}{A}\right)}{P\left(A\right) \cdot P\left(\dfrac{E}{A}\right) + P\left(B\right) \cdot P\left(\dfrac{E}{B}\right) + P\left(C\right) \cdot P\left(\dfrac{E}{C}\right)}$

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

$= \dfrac{28}{45}$

option \left(4\right)$