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14 percent of a certain company's life insurance policy holders are smokers. For each nonsmoker the probability of dying during the year is 0.012. For each smoker the probability of dying during the year is 0.04. Find the probability that a policy holder who died last year was a smoker.

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Answer:

0.3517 is the probability that a policy holder who died last year was a smoker.              

Step-by-step explanation:

We are given the following in the question:

Probability smoker = 14%

[tex]P(S) = 0.14[/tex]

Probability not a smoker =

[tex]P(S') = 1 - 0.14 = 0.86[/tex]

Probability of dying given nonsmoker =

[tex]P(D|S') = 0.012[/tex]

Probability of dying given they are a smoker =

[tex]P(D|S) = 0.04[/tex]

We have to find the probability that a policyholder was a smoker given that they died.

We have to evaluate:

[tex]P(S|D)[/tex]

Using Bayes's Theorem, we can write

[tex]P(S|D) =\dfrac{ P(D|S)*P(S)}{P(D|S)*P(S) + P(D|S')*P(S')}\\\\P(S|D) = \dfrac{0.04(0.14)}{0.04(0.14) + 0.012(0.86)}\\\\P(S|D) = 0.3517[/tex]

0.3517 is the probability that a policy holder who died last year was a smoker.