CBSE 2026 · Region 2 · Set 1 · Q36 · 4 marks
Smoking increases the risk of lung problems.
A study revealed that $\displaystyle 170$ in $\displaystyle 1000$ males who smoke develop lung complications, while $\displaystyle 120$ out of $\displaystyle 1000$ females who smoke develop lung related problems. In a colony, $\displaystyle 50$ people were found to be smokers of which $\displaystyle 30$ are males. A person is selected at random from these $\displaystyle 50$ 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 person selected at random is detected with lung complications. Find the probability that selected person is a female.A person selected at random is not having lung problems, find the probability that the person is a male.
Smoking increases the risk of lung problems.
A study revealed that $\displaystyle 170$ in $\displaystyle 1000$ males who smoke develop lung complications, while $\displaystyle 120$ out of $\displaystyle 1000$ females who smoke develop lung related problems. In a colony, $\displaystyle 50$ people were found to be smokers of which $\displaystyle 30$ are males. A person is selected at random from these $\displaystyle 50$ 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 person selected at random is detected with lung complications. Find the probability that selected person is a female.
A person selected at random is not having lung problems, find the probability that the person is a male.
Marking-scheme solution
Total number of smokers in the colony $\displaystyle =50$,
Number of male smokers $\displaystyle =30$, Number of female smokers $\displaystyle =20$
Let $\displaystyle L$ represent the event that the person is suffering from the lung complications.
(i)
$\displaystyle P(F)=\dfrac{20}{50}=\dfrac{2}{5}$
(ii)
$\displaystyle P\left(L^{\prime} \mid M\right)=1-P(L \mid M)=1-\dfrac{170}{1000}=\dfrac{83}{100}$
(iii)
$\displaystyle P(L \mid M)=\dfrac{170}{1000}=\dfrac{17}{100}$ and $\displaystyle P(L \mid F)=\dfrac{120}{1000}=\dfrac{12}{100}$
Using Baye's Theorem, $\displaystyle P(F \mid L)=\dfrac{P(F) \cdot P(L \mid F)}{P(M) \cdot P(L \mid M)+P(F) \cdot P(L \mid F)}$
$\displaystyle =\dfrac{\frac{2}{5} \cdot \frac{12}{100}}{\frac{3}{5} \cdot \frac{17}{100}+\frac{2}{5} \cdot \frac{12}{100}}$
$\displaystyle =\dfrac{24}{75}$ or $\displaystyle \dfrac{8}{25}$
Here $\displaystyle P\left(L^{\prime} \mid M\right)=\dfrac{83}{100}, P\left(L^{\prime} \mid F\right)=\dfrac{88}{100}$
Using Baye's Theorem, $\displaystyle P\left(M \mid L^{\prime}\right)=\dfrac{P(M) \cdot P\left(L^{\prime} \mid M\right)}{P(M) \cdot P\left(L^{\prime} \mid M\right)+P(F) \cdot P\left(L^{\prime} \mid F\right)}$
$\displaystyle =\dfrac{\frac{3}{5} \cdot \frac{83}{100}}{\frac{3}{5} \cdot \frac{83}{100}+\frac{2}{5} \cdot \frac{88}{100}}$
$\displaystyle =\dfrac{249}{425}$
ProbabilityBayes' TheoremApplycase_studymedium
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CBSE Class 12 Mathematics past-paper question from the 2026board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.