Key Concepts and Formulas

• Bayes' Theorem: This theorem is used to calculate the conditional probability of an event based on prior knowledge of conditions that might be related to the event. For two events $A$ and $E$, it states: $P(A|E) = \frac{P(A) \cdot P(E|A)}{P(E)}$ Here, $P(A|E)$ is the posterior probability of event $A$ given that event $E$ has occurred. $P(A)$ is the prior probability of event $A$, and $P(E|A)$ is the likelihood of event $E$ occurring given that event $A$ has occurred.

• Law of Total Probability: This law is used to find the total probability of an event $E$ when it can occur in conjunction with several mutually exclusive and exhaustive events $A_1, A_2, \ldots, A_n$. It states: $P(E) = P(A_1)P(E|A_1) + P(A_2)P(E|A_2) + \ldots + P(A_n)P(E|A_n)$ When combined, Bayes' Theorem for multiple events is often written as: $P(A_i|E) = \frac{P(A_i) \cdot P(E|A_i)}{\sum_{j=1}^{n} P(A_j)P(E|A_j)}$

Step-by-Step Solution

Step 1: Define Events and Extract Given Information

We begin by clearly defining the events involved and listing the probabilities provided in the problem statement. Let $E$ be the event that a randomly chosen person suffers from the chest disorder.

We can categorize the 400 people in the group into three mutually exclusive and exhaustive groups based on their characteristics:

• $A_1$: The chosen person is a smoker and non-vegetarian.
• $A_2$: The chosen person is a smoker and vegetarian.
• $A_3$: The chosen person is a non-smoker and vegetarian.

Now, we calculate the prior probabilities $P(A_i)$ for each group based on the number of people in each group relative to the total group size (400 people):

• For event $A_1$ (smoker and non-vegetarian): Number of people = 160 $P(A_1) = \frac{160}{400} = \frac{16}{40} = \frac{2}{5}$
• For event $A_2$ (smoker and vegetarian): Number of people = 100 $P(A_2) = \frac{100}{400} = \frac{10}{40} = \frac{1}{4}$
• For event $A_3$ (non-smoker and vegetarian): Number of people = 140 $P(A_3) = \frac{140}{400} = \frac{14}{40} = \frac{7}{20}$ (We can verify that $P(A_1) + P(A_2) + P(A_3) = \frac{2}{5} + \frac{1}{4} + \frac{7}{20} = \frac{8+5+7}{20} = \frac{20}{20} = 1$, confirming these events cover the entire group.)

Next, we extract the conditional probabilities $P(E|A_i)$ for getting the chest disorder for each group:

• Chance of disorder for $A_1$ (smoker and non-vegetarian) = 35% $P(E|A_1) = 0.35 = \frac{35}{100} = \frac{7}{20}$
• Chance of disorder for $A_2$ (smoker and vegetarian) = 20% $P(E|A_2) = 0.20 = \frac{20}{100} = \frac{1}{5}$
• Chance of disorder for $A_3$ (non-smoker and vegetarian) = 10% $P(E|A_3) = 0.10 = \frac{10}{100} = \frac{1}{10}$

Step 2: Calculate the Total Probability of Suffering from the Disorder ($P(E)$)

To apply Bayes' Theorem, we first need to calculate the overall probability that a randomly chosen person from the group suffers from the chest disorder, $P(E)$. We use the Law of Total Probability, summing the products of prior probabilities and their respective conditional probabilities: $P(E) = P(A_1)P(E|A_1) + P(A_2)P(E|A_2) + P(A_3)P(E|A_3)$ Substitute the values from Step 1: $P(E) = \left(\frac{2}{5} \times \frac{7}{20}\right) + \left(\frac{1}{4} \times \frac{1}{5}\right) + \left(\frac{7}{20} \times \frac{1}{10}\right)$ $P(E) = \frac{14}{100} + \frac{1}{20} + \frac{7}{200}$ To sum these fractions, we find a common denominator, which is 200: $P(E) = \frac{14 \times 2}{100 \times 2} + \frac{1 \times 10}{20 \times 10} + \frac{7}{200}$ $P(E) = \frac{28}{200} + \frac{10}{200} + \frac{7}{200}$ $P(E) = \frac{28 + 10 + 7}{200} = \frac{45}{200}$

Step 3: Apply Bayes' Theorem to find the required probability

The problem asks for the probability that the selected person is a non-smoker and vegetarian, given they are suffering from the chest disorder. This corresponds to finding $P(A_3|E)$. Using Bayes' Theorem: $P(A_3|E) = \frac{P(A_3) \cdot P(E|A_3)}{P(E)}$ From our calculations in Step 1, the numerator $P(A_3)P(E|A_3)$ is: $P(A_3)P(E|A_3) = \frac{7}{20} \times \frac{1}{10} = \frac{7}{200}$ Now, substitute this value and the total probability $P(E)$ (calculated in Step 2) into Bayes' Theorem: $P(A_3|E) = \frac{\frac{7}{200}}{\frac{45}{200}}$ $P(A_3|E) = \frac{7}{200} \times \frac{200}{45}$ $P(A_3|E) = \frac{7}{45}$

Common Mistakes & Tips

• Misidentifying the Target Event: Carefully read what probability the question is asking for. Ensure you are calculating $P(A_i|E)$ for the correct $A_i$.
• Errors in Law of Total Probability: A common mistake is to incorrectly calculate the denominator $P(E)$. Ensure all mutually exclusive and exhaustive events are included and their probabilities correctly multiplied and summed.
• Fractional Arithmetic: Be precise with fraction operations, finding common denominators for addition/subtraction, and correctly inverting for division.

Summary

This problem demonstrates a classic application of Bayes' Theorem. We first identified the different groups of people and their prior probabilities, along with their respective chances of having the chest disorder. Then, we calculated the total probability of a person having the disorder using the Law of Total Probability. Finally, we applied Bayes' Theorem to find the conditional probability that a person is a non-smoker and vegetarian, given they suffer from the chest disorder, which came out to be $\frac{7}{45}$.

The final answer is $\boxed{\frac{7}{45}}$, which corresponds to option (C).