A large research hospital has accumulated statistical data onits patients for an

Question

A large research hospital has accumulated statistical data onits patients for an extended period. Researchers havedetermined that patients who are smokers have an 18% chance ofcontracting a serious illness such as heart disease, cancer, oremphysema, whereas there is only a .06 probability that a nonsmokerwill contract a serious illness. From hospital records, theresearchers know that 23% of all hospital patients are smokers,while 77% are nonsmokers. For planning purposes, the hospitalphysician staff would like to know the probability that a givenpatient is a smoker if the patient has a serious illness.

Explanation / Answer

S=smoker;   S'=non smoker I=ill; I'=not ill We are given the following values: P(I/S)=.18; P(I/S')=.06 P(S)=.23; P(S')=.77 Find P(S/I). P(S/I)=P(SnI)/P(I)=P(I/S)P(S)/P(I) =(.18)(.23)/P(I)=.0414/P(I) P(I)=P(SnI)+P(S'nI) =P(I/S)P(S)+P(I/S')P(S') =.0414+(.06)(.77) P(I)=.0876 Therefore, P(S/I)=.0414/(.0876)=.4726

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