Search any question & find its solution
Question:
Answered & Verified by Expert
A survey of people in a given region showed that $20 \%$ were smokers. The probability of death due to lung cancer given that a person smoked was 10 times the probability of death due to lung cancer, given that a person did not smoke. If the probability of death due to lung cancer in the region is 0.006 . What is the probability of death due to lung cancer given that a person is a smoker?
Options:
Solution:
1598 Upvotes
Verified Answer
The correct answer is:
$3 / 140$
Let $S=$ Event that person is smoker
$\mathrm{NS}=$ Event that person is non-smoker
$D=$ Event that death is due to lung cancer.
Now, probability of death due to cancer that a person is a smoker, $P(D)=P(S) \cdot P\left(\frac{D}{S}\right)+P(N S) \cdot P\left(\frac{D}{N S}\right)$
$\Rightarrow 0.006=\frac{20}{100} \times P\left(\frac{D}{S}\right)+\frac{80}{100} \times \frac{1}{10} \times P\left(\frac{D}{S}\right)$
$\Rightarrow \frac{6}{1000}=\frac{2}{10} P\left(\frac{D}{S}\right)+\frac{8}{100} P\left(\frac{D}{S}\right)$
$\Rightarrow \quad P\left(\frac{D}{S}\right)\left[\frac{2}{10}+\frac{8}{100}\right]=\frac{6}{1000}$
$\Rightarrow \quad P\left(\frac{D}{S}\right)\left[\frac{20+8}{100}\right]=\frac{6}{1000}$
$\therefore \quad P\left(\frac{D}{S}\right)=\frac{6}{1000} \times \frac{100}{28}$
$=\frac{3}{140}$
$\mathrm{NS}=$ Event that person is non-smoker
$D=$ Event that death is due to lung cancer.
Now, probability of death due to cancer that a person is a smoker, $P(D)=P(S) \cdot P\left(\frac{D}{S}\right)+P(N S) \cdot P\left(\frac{D}{N S}\right)$
$\Rightarrow 0.006=\frac{20}{100} \times P\left(\frac{D}{S}\right)+\frac{80}{100} \times \frac{1}{10} \times P\left(\frac{D}{S}\right)$
$\Rightarrow \frac{6}{1000}=\frac{2}{10} P\left(\frac{D}{S}\right)+\frac{8}{100} P\left(\frac{D}{S}\right)$
$\Rightarrow \quad P\left(\frac{D}{S}\right)\left[\frac{2}{10}+\frac{8}{100}\right]=\frac{6}{1000}$
$\Rightarrow \quad P\left(\frac{D}{S}\right)\left[\frac{20+8}{100}\right]=\frac{6}{1000}$
$\therefore \quad P\left(\frac{D}{S}\right)=\frac{6}{1000} \times \frac{100}{28}$
$=\frac{3}{140}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.