According to a survey of American households, the probability that the residents own two cars if annual household income is over $50,000 is 80%. Of the households surveyed, 60% had incomes over $50,000 and 70% had two cars. The probability that annual household income is over $50,000 if the residents of a household do not own two cars is ________.
A) 0.12
B) 0.18
C) 0.40
D) 0.70

Question

Quizplus · Accepted Answer

We can use Bayes' theorem to solve this problem:

P(income>50k | not 2 cars) = P(not 2 cars | income>50k) * P(income>50k) / P(not 2 cars)

We are given P(not 2 cars | income>50k) = 0.20 (because the complement of 80% is 20%), P(income>50k) = 0.60 (60% of households have income over $50k), and P(not 2 cars) = 0.30 (because the complement of 70% is 30%).

Plugging in these values, we get:

P(income>50k | not 2 cars) = 0.20 * 0.60 / 0.30 = 0.40

Therefore, the probability that annual household income is over $50,000 if the residents of a household do not own two cars is 0.40 or 40%. The best choice is (C).

According to a Survey of American Households, the Probability That