Statistics and Probability: Mathematics | JEE Main
25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is 10k. Then the value of k is.
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Hint 1 of 3
Let S be the event that a person is a smoker, S′ be a non-smoker, and C be the event that a person has lung cancer. If P(C∣S′)=p, what are the prior probabilities and conditional probabilities in terms of p?
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Correct answer
The value of k is 9.
Option analysis
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Solution
StepWorking
01given
Let S be the event that a person is a smoker and NS that the person is a non-smoker. P(S)=25%=41, P(NS)=75%=43. Let C be the event of developing lung cancer. According to the question, P(C∣S)=27P(C∣NS).
02goal
Find the value of k given that the posterior probability P(S∣C)=10k.
03approach
Use Bayes' Theorem to calculate the posterior probability: P(S∣C)=P(S)P(C∣S)+P(NS)P(C∣NS)P(S)P(C∣S). Then substitute P(C∣S)=27P(C∣NS) so that the unknown base rate P(C∣NS) cancels out.
Check with population numbers: Out of 100 people, 25 are smokers and 75 are non-smokers. If the cancer rate for non-smokers is p, cancer cases among non-smokers is 75p, and among smokers is 25×27p=675p. Total cancer cases =750p. Fraction of smokers =675p/750p=675/750=9/10. Thus, k=9.
Hints that build this answer step by step
Let S be the event that a person is a smoker, S′ be a non-smoker, and C be the event that a person has lung cancer. If P(C∣S′)=p, what are the prior probabilities and conditional probabilities in terms of p?
P(S)=41, P(S′)=43, P(C∣S′)=p, P(C∣S)=27p
Which formula correctly gives the posterior probability P(S∣C) using Bayes' Theorem?
P(S∣C)=P(S)P(C∣S)+P(S′)P(C∣S′)P(S)P(C∣S)
Substituting the probabilities into the formula, what is the value of P(S∣C) and consequently k?
Does '27 times more chances' mean 27 times as likely or (1 + 27) times?
In standard Indian competitive exams like JEE, 'n times more' often means 'n times as much'. Therefore, the likelihood ratio is taken directly as n. Here, that value is 27.