According to a survey of Australian households,the probability that the residents own 2 cars if annual household income is over $50,000 is 80%.Of the households surveyed,60% had incomes over $50,000 and 70% had 2 cars.The probability that the residents do not own 2 cars if annual household income is not over $50,000 is
A)0.18.
B)0.45.
C)0.12.
D)0.70.

Question

Quizplus · Accepted Answer

Let's use the following events: 
A: annual income over $50,000 
B: household owns 2 cars

We know:

P(B|A) = 0.8 
P(A) = 0.6 
P(B) = 0.7

We want to find:

P(not B|not A)

Using Bayes' theorem, we have:

P(not B|not A) = P(not A|not B) * P(not B) / P(not A)

We can find P(not A|not B) using the fact that:

P(not A|not B) = 1 - P(A|not B)

P(A|not B) = P(A and B) / P(not B) = (1 - P(not A)) / P(not B)

P(not A) = 1 - P(A) = 0.4

P(A and B) = P(B|A) * P(A) = 0.8 * 0.6 = 0.48

Therefore:

P(A|not B) = (1 - 0.48) / (1 - 0.7) = 0.4

P(not B|not A) = (1 - 0.4) * (1 - 0.7) / 0.4 = 0.45

So the answer is (B).

According to a Survey of Australian Households,the Probability That the Residents