According to a survey of Australian households, the probability that annual household income is over $50,000 if the residents own 2 cars is 80%. Of the households surveyed, 60% had incomes over $50,000 and 70% had 2 cars. The probability that annual household income is over $50,000 if the residents of a household do not own 2 cars is?
0.48
To find the probability that annual household income is over $50,000 given that the residents do not own 2 cars, we can make use of conditional probability.
Let's denote the event "annual household income over $50,000" as A and the event "residents own 2 cars" as B. We are given the following information:
P(A|B) = 0.8 (probability of income over $50,000 if residents own 2 cars)
P(A) = 0.6 (probability of income over $50,000 overall)
P(B) = 0.7 (probability of residents owning 2 cars)
We want to find P(A|not B), i.e., the probability of income over $50,000 given that residents do not own 2 cars.
To find this, we need to use Bayes' theorem, which states:
P(A|not B) = [P(not B|A) * P(A)] / [P(not B|A) * P(A) + P(not B|not A) * P(not A)]
P(not B|A) represents the probability of not owning 2 cars given that annual household income is over $50,000. We are not given this information, so we cannot directly calculate it. However, we can take the complement of P(B|A) and use it instead, since it is equal to 1 - P(not B|A).
Let's denote P(not B|A) as C. Therefore:
C = 1 - P(B|A) = 1 - 0.8 = 0.2
Now, we need to find P(not B|not A), which is the probability of not owning 2 cars given that annual household income is not over $50,000. To calculate this, we can subtract P(B|not A) from the complement of 1.
P(B|not A) represents the probability of owning 2 cars given that annual household income is not over $50,000.
Let's denote P(B|not A) as D, then:
D = 1 - P(not B|not A)
We are not given this information either, so we need to make an assumption based on what we know so far. Since we know that 70% of the households surveyed have 2 cars, and the probability of income over $50,000 is 60% overall, we can assume that the percentage of households with an income below $50,000 is 40%.
Therefore:
P(not B|not A) = 1 - D = 1 - 0.4 = 0.6
Now, we can substitute the values into Bayes' theorem:
P(A|not B) = [C * P(A)] / [C * P(A) + D * P(not A)]
= [0.2 * 0.6] / [0.2 * 0.6 + 0.6 * 0.4]
= 0.12 / (0.12 + 0.24)
= 0.12 / 0.36
= 0.3333...
Therefore, the probability that annual household income is over $50,000 given that the residents do not own 2 cars is approximately 0.3333, or 33.33%.
To find the probability that annual household income is over $50,000 if the residents of a household do not own 2 cars, we need to use the information given in the survey.
Let's break down the given information:
- Probability of annual household income being over $50,000 given that the residents own 2 cars: 80%
- Percentage of households surveyed that had annual incomes over $50,000: 60%
- Percentage of households surveyed that had 2 cars: 70%
To find the probability that annual household income is over $50,000 if the residents do not own 2 cars, we need to subtract the households that have 2 cars from the total households surveyed.
Percentage of households without 2 cars = 100% - Percentage of households with 2 cars = 100% - 70% = 30%
Now, we can calculate the probability using the formula:
Probability (annual household income > $50,000 | not 2 cars) = Probability (annual household income > $50,000 and not 2 cars) / Probability (not 2 cars)
Let's calculate the numerator:
Probability (annual household income > $50,000 and not 2 cars) = Probability (annual household income > $50,000) - Probability (annual household income > $50,000 and 2 cars)
Probability (annual household income > $50,000 and 2 cars) = Probability (annual household income > $50,000 | 2 cars) * Probability (2 cars)
Probability (annual household income > $50,000 and 2 cars) = 0.8 * 0.7 = 0.56
Probability (annual household income > $50,000) = 0.6
Therefore,
Probability (annual household income > $50,000 and not 2 cars) = 0.6 - 0.56 = 0.04
Now, let's calculate the probability of not having 2 cars:
Probability (not 2 cars) = 30% = 0.3
Finally, we can calculate the probability we were looking for:
Probability (annual household income > $50,000 | not 2 cars) = 0.04 / 0.3 ≈ 0.1333 or 13.33%
The probability that annual household income is over $50,000 if the residents of a household do not own 2 cars is approximately 13.33%.