Question

Smoking increases the risk of lung problems. A study revealed that 170 in 1000 males who smoke develop lung complications, while 120 out of 1000 females who smoke develop lung related problems. In a colony, 50 people were found to be smokers of which 30 are males. A person is selected at random from these 50 people and tested for lung related problems. Based on the given information answer the following questions: 

(i) What is the probability that selected person is a female? 
(ii) If a male person is selected, what is the probability that he will not be suffering from lung problems? 
(iii)(a) A person selected at random is detected with lung complications. Find the probability that selected person is a female. 
OR 
(iii)(b) A person selected at random is not having lung problems. Find the probability that the person is a male. 
 

Hint:

When conditional probabilities are given, Bayes’ theorem is useful to reverse the conditioning. Always convert percentages or ratios into probabilities before applying the formula.

Answer 1

(i) What is the probability that the selected person is a female? 
The probability that the selected person is a female is given by the ratio of the number of females to the total number of smokers: \[ P(\text{female}) = \frac{\text{Number of females}}{\text{Total number of smokers}} = \frac{20}{50} = \frac{2}{5} \] So, the probability that the selected person is a female is \( \frac{2}{5} \). 

(ii) If a male person is selected, what is the probability that he will not be suffering from lung problems? 
We are given that out of 1000 males who smoke, 170 develop lung problems. Therefore, the number of males who do not develop lung problems is: \[ 1000 - 170 = 830 \] The probability that a randomly selected male does not have lung problems is: \[ P(\text{no lung problems | male}) = \frac{830}{1000} = 0.83 \] So, the probability that a selected male will not be suffering from lung problems is \( 0.83 \). 

(iii)(a) A person selected at random is detected with lung complications. Find the probability that the selected person is a female. 
To find the probability that the selected person is a female given that they have lung complications, we use Bayes' Theorem: \[ P(\text{female | lung problems}) = \frac{P(\text{lung problems | female}) \cdot P(\text{female})}{P(\text{lung problems})} \] Where: - \( P(\text{female}) = \frac{20}{50} = \frac{2}{5} \), - \( P(\text{lung problems | female}) = \frac{120}{1000} \), - \( P(\text{lung problems}) \) is the total probability of having lung problems: \[ P(\text{lung problems}) = P(\text{lung problems | male}) \cdot P(\text{male}) + P(\text{lung problems | female}) \cdot P(\text{female}) \] \[ = \frac{170}{1000} \cdot \frac{30}{50} + \frac{120}{1000} \cdot \frac{20}{50} \] \[ = \frac{170}{1000} \cdot \frac{3}{5} + \frac{120}{1000} \cdot \frac{2}{5} = 0.15 \] Now applying Bayes' Theorem: \[ P(\text{female | lung problems}) = \frac{\frac{120}{1000} \cdot \frac{2}{5}}{0.15} = \frac{\frac{240}{5000}}{0.15} = \frac{240}{750} = \frac{8}{25} \] So, the probability that the selected person is a female, given that they have lung complications, is \( \frac{8}{25} \). 

(iii)(b) A person selected at random is not having lung problems. Find the probability that the person is a male. 
The probability that a person has no lung problems is: \[ P(\text{no lung problems}) = 1 - P(\text{lung problems}) = 1 - 0.15 = 0.85 \] We are asked to find the probability that the selected person is male, given that they don't have lung problems. Using Bayes' Theorem: \[ P(\text{male | no lung problems}) = \frac{P(\text{no lung problems | male}) \cdot P(\text{male})}{P(\text{no lung problems})} \] Where: - \( P(\text{no lung problems | male}) = 1 - \frac{170}{1000} = 0.83 \), - \( P(\text{male}) = \frac{30}{50} = \frac{3}{5} \), Now calculate \( P(\text{no lung problems}) \): \[ P(\text{no lung problems}) = \frac{830}{1000} \cdot \frac{30}{50} + \frac{880}{1000} \cdot \frac{20}{50} = 0.85 \] Now applying Bayes' Theorem: \[ P(\text{male | no lung problems}) = \frac{0.83 \cdot \frac{3}{5}}{0.85} = \frac{0.498}{0.85} = \frac{249}{425} \] So, the probability that the selected person is male, given that they don't have lung problems, is \( \frac{249}{425} \). 

Final Answers: 
1. The probability that the selected person is a female is \( \frac{2}{5} \). 
2. If a male person is selected, the probability that he will not be suffering from lung problems is \( 0.83 \). 
3. If a person selected at random is detected with lung complications, the probability that the selected person is a female is \( \frac{8}{25} \). 
4. If a person selected at random is not having lung problems, the probability that the person is male is \( \frac{249}{425} \).