Question

According to a survey of American households, the probability that residents own 2 cars if the annual household income is over $25,000 is 80%. Among the households surveyed, 60% had incomes over $25,000, and 70% had 2 cars. Your task is to find the probability that the annual household income is over $25,000 if the residents of a household own 2 cars.

Which of the following options represents this probability?

(A) 0.42
(B) 0.48
(C) 0.50
(D) 0.69

Please calculate and select the correct probability.

Answer 1

The probability that the annual household income is over $25,000 if the residents of a household own 2 cars is approximately 0.686.

Step-by-step explanation:

To find the probability that the annual household income is over $25,000 if the residents of a household own 2 cars, we can use Bayes' theorem. Bayes' theorem states that P(A|B) = (P(B|A) * P(A)) / P(B), where A and B are events. In this case, A represents the event of owning 2 cars and B represents the event of having an annual household income over $25,000. Given that the probability of owning 2 cars if the annual household income is over $25,000 is 80% (P(A|B) = 0.8), the probability of having an annual household income over $25,000 is 60% (P(B) = 0.6), and the overall probability of owning 2 cars is 70% (P(A) = 0.7), we can substitute these values into the formula:

P(B|A) = (P(A|B) * P(B)) / P(A)

P(B|A) = (0.8 * 0.6) / 0.7 = 0.686

Therefore, the probability that the annual household income is over $25,000 if the residents of a household own 2 cars is approximately 0.686.