Question

According to a survey of American households, the probability that the residents own 2 cars if annual household income is over $25,000 is 80%. Of the households surveyed, 60% had incomes over $25,000 and 70% had 2 cars. Use the Bayes Theorem to calculate the probability that annual household income is at most $25,000 if the residents of a household do not own 2 cars.

Answer 1

P(not have 2 cars )=1-P(2 cars )=1-0.70 =0.3

P(income at most 25000) =1-0.60=0.4

P(not have 2 cars and income over 25000) =P( income over 25000)*P(not have 2 cars| income over 25000) =0.6*(1-0.80) =0.12

P(not have 2 cars and income at most $25,000 )=P(not have 2 cars)-P(not have 2 cars and income over 25000) =0.3-0.12 =0.18

therefore P(income at most $25,000 given do not own 2 cars )

=P(not have 2 cars and income at most $25,000 )/P(not have 2 cars )

=0.18/0.3 =0.60