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 chances 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 :

Solution

Maths sir · 2 steps

Count how many people in each group actually have the disorder, then ask what share is smoker non-veg.

1. Set up the events. Let E₁, E₂, E₃ be the three groups and A the event that the person has the disorder. We need P(E₁ | A), the chance of the group given the disorder, so this is Bayes' theorem.

   Group	People	Chance of disorder	People with disorder
   E₁: smoker, non-veg	160	35%	160 × 0.35 = 56
   E₂: smoker, veg	100	20%	100 × 0.20 = 20
   E₃: non-smoker, veg	140	10%	140 × 0.10 = 14
   Total	400	90/400 = 22.5% overall	56 + 20 + 14 = 90

   P(E₁ | A) = P(E₁)P(A | E₁)/(Σᵢ₌₁³ P(Eᵢ)P(A | Eᵢ)) = 0.14/(0.14 + 0.05 + 0.035) = 56/90

2. So. Of the 90 people with the disorder, 56 are smokers and non-vegetarian, so the probability is 56/90 = 28/45, option (d).

The idea to keep. In Bayes' problems, turn each group into the number of people who show the symptom. The answer is your group's share of that total.

Answer from the NTA final answer key. Solution by PrepWiser.