JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Probability

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25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer then a non-smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is

\frac{k}{10}

. Then the value of k is ______.

Answer: 9

Solution

Let smoking and non-smoking carry prior probabilities

\frac{1}{4}

and

\frac{3}{4}

, and let a smoker's chance of lung cancer be

27

times a non-smoker's, say

27 p

against

p

. By Bayes' theorem the probability that a diagnosed person is a smoker is

\frac{\frac{1}{4} \cdot 27 p}{\frac{1}{4} \cdot 27 p + \frac{3}{4} \cdot p} = \frac{27}{27 + 3} = \frac{27}{30} = \frac{9}{10}

. Comparing with

\frac{k}{10}

gives

k = 9

JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Integer answer

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