Consumer surveys indicate that 40% of newspaper readers read automobile ads and

Question

Consumer surveys indicate that 40% of newspaper readers read automobile ads and 5% of those who read the ads actually purchase automobiles. On the other hand, 50% of magazine readers read automobile ads but only 3% of those who read the ads actually buy a car. Among those who do not read either newspaper or magazine auto ads, 1% buy cars anyway. Sixty percent of the population reads newspapers, while 20 percent primarily read magazines. Compute the overall percentage of the population that purchase automobiles in a given year.

Explanation / Answer

According to the question P(Newspaper readers)=0.6 (60% of the population reads newspaper) P(Magazine readers but not newspaper readers)=0.2 (20% primarily read magazines) According to total probability theorem P(Newspaper readers)+P(magazine not newspaper readers)+P(neither newspaper nor magazine readers) = 1 P(neither newspaper nor magazine)= 1-0.6-0.2= 0.2 P(doesn't read auto ads)= P(doesn't read auto ads|newspaper reader)*P(newspaper reader) + P(doesn't read auto ads|primarily magazine reader)*P(primarily magazine reader) + P(doesn't read either NP or magazine) P(doesn't read auto ads in NP|reads newspaper)= 1 - P(reads auto ads|reads newspaper) = 1 - 0.4 = 0.6 P(doesn't read auto ads in magazines|reads primarily magazines)= 1 - P(reads auto ads|reads primarily magazines) = 1 - 0.5 = 0.5 P(doesn't read auto ads) = 0.6*0.6 + 0.2*0.5 + 0.2 = 0.36+0.10+0.2 = 0.66 P(auto purchases)= P(purchases|read auto ads in NP)*P(read auto ads|newspaper readers)*P(newspaper readers) + P(purchases auto|read auto ads in magazine)*P(read auto ads|read magazines)*P(reads primarily magazines) + P(purchases auto|doesn't read newspaper or magazines auto ads)*P(doesn't read auto ads) P(auto purchases) = 0.05*0.4*0.6 + 0.03*0.5*0.2 + 0.01*0.66 = .0216 Overall percentage of population that purchases automobiles in a given year= 2.16%

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