A study is conducted over 5 years regarding the death of smokers.At the begining

Question

A study is conducted over 5 years regarding the death of smokers.At the begining of the study, 20% were classified as heavy smokers,30% were classified as light smokers, and 50% were classified asnon-smokers. At the end of the study, it was determined that lightsmokers were twice as likely to die than non-smokers, but only halfas likely as heavy smokers. Given a randomly selected person hasdied during the 5 year study, what is the probability that theperson was a heavy smoker?
**PLEASE EXPLAIN** The problem i have here, is that it does not give any actual numberfor any of the outcome deaths. If it gave any of the 3, i couldfigure out the rest cuz it scales by 2 from non-smokers -> lightsmokers -> heavy smokers, but since it does not give an actualnumber for any of the 3, i cannot seem to figure out how to do thisproblem. Any help is appreciated A study is conducted over 5 years regarding the death of smokers.At the begining of the study, 20% were classified as heavy smokers,30% were classified as light smokers, and 50% were classified asnon-smokers. At the end of the study, it was determined that lightsmokers were twice as likely to die than non-smokers, but only halfas likely as heavy smokers. Given a randomly selected person hasdied during the 5 year study, what is the probability that theperson was a heavy smoker?
**PLEASE EXPLAIN** The problem i have here, is that it does not give any actual numberfor any of the outcome deaths. If it gave any of the 3, i couldfigure out the rest cuz it scales by 2 from non-smokers -> lightsmokers -> heavy smokers, but since it does not give an actualnumber for any of the 3, i cannot seem to figure out how to do thisproblem. Any help is appreciated

Explanation / Answer

We are given: P(H)=.2; P(L)=.3; P(N)=.5 Also given the relationships: P(D/L)=2P(D/N); P(D/L)=.5P(D/H)......................(1) We wish to find P(H/D)=P(HnD)/P(D)=P(D/H)P(H)/P(D)....................................(2) Now we find P(D)=P(DnH)+P(DnL)+P(DnN) =P(D/H)P(H)+P(D/L)P(L)+P(D/N)P(N) Given the relationships in (1) above we can write theexpression in terms of P(D/L). We have, P(D)=2P(D/L)(.2)+P(D/L)(.3)+(.5)P(D/L)(.5)=.95P(D/L)=.95P(D/L) Using (1) we have P(D)=.95P(D/L)=.95(.5)P(D/H)=.475P(D/H) Now substitute into (2) and noting that the P(D/H) cancel weget P(H/D)=P(H)/.475=.2/.475=.421

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