Question

An insurer knows that the proportion of smokers in a population is 0.3. Furthermore, it is known that: 40% of applicants who are smokers say they do not smoke on their applications. None of the non-smoking applicants are lying on their application.
Calculate what is the probability that one of the applicants says that he does not smoke, and that in fact do not smoke.

Answer 1

Answer: Approximately 0.8537

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Explanation:

Let's say this population had 1000 people. You can pick any number you want, but I'm going for some nice big round number to work with.

The proportion of smokers is 0.3, which means 0.3*1000 = 300 people in this population are smokers and 1000-300 = 700 are nonsmokers.

Of the 300 people who do smoke, 40% of them lie on the application. This means 0.40*300 = 120 people lied. These 120 people say they don't smoke, but in fact they do smoke.

So if we were to survey this population, then 700+120 = 820 people would say they don't smoke

Only considering the people who write "don't smoke" on the survey, which is that 820 figure, 700 of them are true nonsmokers. Therefore, the probability we want is 700/820 = 0.8537 approximately