Ill be concise:60% of people in survey had income greater than 25k, 40% under 25k. Further, same survey, 70% have exactly 2 cars, while 30% have a different number than 2. The probability that the residents own 2 cars IF income is over 25k is 80%.

Question

Ill be concise:60% of people in survey had income greater than 25k, 40% under 25k. Further, same survey, 70% have exactly 2 cars, while 30% have a different number than 2. The probability that the residents own 2 cars IF income is over 25k is 80%.

what is the probability of a household have exactly 2 cars AND an income UNDER 25k?
i get the answer 28%. the right answer is 22%.
I don't know how to incorporate the last 80% chance bit of the setup part. please help

Answer 1

To find the probability of a household having exactly 2 cars and an income under 25k, we need to calculate the intersection of these two events. Let's break it down step by step:

1. First, let's find the probability that a household has an income under 25k and the probability that they own exactly 2 cars separately.

The probability of an income under 25k is given as 40%.
The probability of owning exactly 2 cars is given as 70%.

So, the probability of a household having an income under 25k is 0.40, and the probability of owning exactly 2 cars is 0.70.

2. Now, let's consider the additional information given: the probability that residents own 2 cars if their income is over 25k is 80%.

This means that if a household's income is over 25k, the probability of them owning exactly 2 cars is 0.80.

3. To find the probability of a household having exactly 2 cars and an income under 25k, we need to calculate the intersection of these two events.

We can use conditional probability to incorporate the information given. Let's assume the event A represents an income over 25k, and B represents owning exactly 2 cars.

The probability of A (income over 25k) is 0.60, and the probability of B (owning exactly 2 cars) is 0.70.

The probability of owning exactly 2 cars given an income over 25k is 0.80.

Using the formula for conditional probability:
P(A ∩ B) = P(A) × P(B|A)

P(A ∩ B) = (0.60) × (0.80) = 0.48

So, the probability of a household having exactly 2 cars and an income above 25k is 0.48.

4. Finally, to find the probability of a household having exactly 2 cars and an income under 25k:

P(A' ∩ B) = P(B) - P(A ∩ B)

P(A' ∩ B) = 0.70 - 0.48 = 0.22

Thus, the probability of a household having exactly 2 cars and an income under 25k is 0.22, or 22%.

Note: It's important to carefully interpret and use the given information in a problem to arrive at the correct solution.