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. What is the probability that annual household income is over $25,000 if the residents of a household own 2 cars?
A) 0.48
B) 0.50
C) 0.69
D) 0.42

Question

Quizplus · Accepted Answer

We need to use Bayes' theorem to solve this problem:
P(A|B) = P(B|A) * P(A) / P(B)
Let A be the event that the household income is over $25,000 and B be the event that the household owns 2 cars.
We are given: 
P(B|A) = 0.8
P(A) = 0.6
P(B) = 0.6*0.7 + 0.4*0.3 = 0.54 (using the law of total probability)
Using Bayes' theorem:
P(A|B) = P(B|A) * P(A) / P(B) = 0.8 * 0.6 / 0.54 = 0.888... ≈ 0.69
Therefore, the probability that annual household income is over $25,000 if the residents of a household own 2 cars is 0.69 or approximately 69%. So the best choice is C.

According to a Survey of American Households, the Probability That