In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. Their chanced of getting a particular chest disorder are 35%, 20% and 10% respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non - vegetarian is

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$\frac{28}{45}$

$\frac{7}{45}$

$\frac{8}{45}$

$\frac{14}{45}$

answer is A.

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Detailed Solution

$E_{1} =$ People are Smokers and non-veg

$E_{2} =$ People are Smokers and veg

$E_{3} =$ People are non-Smokers and veg

$\begin{array}{rcl} P (E_{1}) & = & \frac{160}{400} P (E_{2}) = \frac{100}{400} P (E_{3}) = \frac{140}{400} \\ = & \frac{16}{40} = \frac{10}{40} = \frac{14}{40} \end{array}$

A=Person Suffering from Chest disorder

$\begin{array}{rcl} P (\frac{A}{E_{1}}) & = & \frac{35}{100} P (\frac{A}{E_{2}}) = \frac{20}{100} P (\frac{A}{E_{3}}) = \frac{10}{100} \\ P (\frac{A}{E_{1}}) & = & \frac{\frac{16}{40} \times \frac{35}{100}}{\frac{16}{40} \times \frac{35}{100} + \frac{10}{40} \times \frac{20}{100} + \frac{14}{40} \times \frac{10}{100}} \\ = & \frac{28}{45} \end{array}$