Problem 42 - A First Course in Probability

Question

Twelve percent of all U.S. households are in California. A total of 1.3 percent of all U.S. households earn more than \(\$ 250,000\) per year, while a total of 3.3 percent of all California households earn more than \(\$ 250,000\) per year. (a) What proportion of all non-California households earn more than \(\$ 250,000\) per year? (b) Given that a randomly chosen U.S. household earns more than \(\$ 250,000\) per year, what is the probability it is a California household?

Step-by-Step Solution

Answer

(a) The proportion of all non-California households earning more than $250,000 per year is approximately 1.02%. (b) The probability that a randomly chosen U.S. household earning more than $250,000 per year is a California household is approximately 30.5%.

Step 1: (a) Proportion of Non-California households earning more than \(250,000 per year

To find the proportion of non-California households that earn more than \)250,000 per year, we will first find the proportion of U.S. households that are not in California. Then, we'll calculate the proportion of all U.S. households earning more than \(250,000 per year and subtract the proportion of California households earning more than \)250,000 per year from it. Finally, we'll divide this value by the proportion of non-California households. 1. Find the proportion of non-California households: The problem states that 12% of all U.S. households are in California, so 100% - 12% = 88% of the households are non-California. That is \(0.88\) proportion of non-California households. 2. Calculate the overall proportion of households earning more than $250,000 per year, which is given as 1.3% or \(0.013\) proportion. 3. Calculate the proportion of California households earning more than $250,000 per year, which is given as 3.3% or \(0.033\) proportion. 4. Calculate the proportion of all U.S. households earning more than $250,000 per year that are non-California households: \(0.013 - (0.12\times0.033)=0.013-0.00396\) 5. Divide the proportion of non-California households earning more than $250,000 per year by the proportion of non-California households: \(\frac{0.013-0.00396}{0.88} = 0.010236\) So the proportion of all non-California households earning more than $250,000 per year is approximately 1.02%.

Step 2: (b) Probability that a randomly chosen high-earning household is a California household

To find the probability that a randomly chosen U.S. household earning more than \(250,000 per year is a California household, we will divide the proportion of California households earning more than \)250,000 per year by the overall proportion of households earning more than $250,000 per year. 1. Calculate the proportion of California households earning more than $250,000 per year, which is given as 3.3% or \(0.033\) proportion. 2. Calculate the overall proportion of households earning more than $250,000 per year, which is given as 1.3% or \(0.013\) proportion. 3. Divide the proportion of California households earning more than \(250,000 per year by the overall proportion of households earning more than \)250,000 per year: \(\frac{0.033\times0.12}{0.013} \approx 0.305\) So the probability that a randomly chosen U.S. household earning more than $250,000 per year is a California household is approximately 30.5%.

Key Concepts

Conditional Probability

Conditional probability is an essential concept in statistics and probability theory. It refers to the likelihood of an event occurring given that another event has already happened. In the context of the exercise, we are interested in finding out the probability that a random household earning more than \( \\(250,000 \) is from California.

Here's how conditional probability works in simple terms:

• We start with a known condition (a household earns more than \( \\)250,000 \)).
• We then calculate the probability of the event we are interested in (that this household is from California).
• The known condition helps us narrow down our sample space to only those households that meet the criterion.

To solve part (b) of the original exercise, we use the formula for conditional probability:\[P(A \mid B) = \frac{P(A \cap B)}{P(B)}\]- Here, \( A \) is the event that a household is from California, and \( B \) is the event that a household earns more than \( \\(250,000 \).- What we did in the solution is essentially calculate \( P(A \cap B) \), the proportion of California households earning more than \( \\)250,000 \), and divide it by \( P(B) \) to find the desired probability.- The conclusion, around 30.5%, tells us a significant proportion of high-earning households in the U.S. are from California when considering those earning over \( \$250,000 \).

Proportion

The concept of proportion is simple yet powerful in understanding distributions and relationships between parts and wholes. It helps us comprehend different parts of a dataset in comparison to the whole set. Let's see how this applies to our exercise.

In the exercise, proportions are used to understand the distribution of high-earning households across different regions:

• Firstly, we determine the portion of all U.S. households that are in California, which is 12%. Therefore, 88% of them are not.
• We also know that 1.3% of all households in the U.S. earn more than \( \\(250,000 \), and that among Californian households, 3.3% earn more than this amount.

By understanding these proportions, we can derive further insights, such as the percentage of non-Californian households earning over \( \\)250,000 \).

For part (a), we adjusted the total proportion of high-earning households to reflect those not in California and then divided by the non-Californian part of the population. The result, about 1.02%, represents non-Californian high earners in the larger set of U.S. households. Proportions like these are vital as they allow us to verify where concentration and distribution are highest, informing economic and demographic analyses.

Household Income Distribution

Household income distribution investigates how household incomes in a given population are spread out across various incomes. This distribution offers insights into economic conditions and inequality in a region.

Let's break down the key points reflected in our example:

• The U.S. household income distribution reveals that a small percentage, 1.3%, earns more than \( \$250,000 \) annually.
• Within California, this figure jumps to 3.3%, suggesting a higher concentration of wealth within the state compared to the national level.

These figures signal several things:- **Economic Power:** California might have better opportunities, industries, or higher living costs driving these earnings.- **Inequality:** When only a small portion earns high incomes, it can highlight income inequality.

Understanding the distribution of household income can lead to broader discussions on taxation, social services, and economic policies. For instance, if a larger proportion of high income is found in a particular region, it may affect budget allocations or tax strategies in that area. In the case of California, this understanding might inform policies tailored to capitalize on or address the high-income concentration.