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Chapter 5: Problem 68

Income tax returns Here is the distribution of the adjusted gross income (in thousands of dollars) reported on individual federal income tax returns in a recent year: $$\begin{array}{lcllll}\hline \text { Income: } & <25 & 25-49 & 50-99 & 100-499 & \geq 500 \\ \text { Probability: } & 0.431 & 0.248 & 0.215 & 0.100 & 0.006 \\\\\hline\end{array}$$ (a) What is the probability that a randomly chosen return shows an adjusted gross income of \(\$ 50,000\) or more? (b) Given that a return shows an income of at least 50,000,what is the conditional probability that the income is at least 100,000 ?

Short Answer

Expert verified
(a) 0.321; (b) 0.330

Step by step solution

01

Identify Relevant Probabilities for Part (a)

To find the probability that a return shows an adjusted gross income of \(50,000 or more, identify the probabilities of the income brackets \)50-99\(, \)100-499\(, and \)\geq 500$, which are 0.215, 0.100, and 0.006 respectively.
02

Calculate Probability for Income $\geq 50,000$

Add up the probabilities identified in Step 1. The probability that a randomly chosen return shows an adjusted gross income of $50,000 or more is calculated as follows: \[ P(\geq 50,000) = 0.215 + 0.100 + 0.006 = 0.321 \]
03

Identify Relevant Probabilities for Part (b)

For the conditional probability, we focus on the income brackets \(50-99\), \(100-499\), and \(\geq 500\) again, but this time to find the probability of being at least \(100,000\) within these brackets.
04

Calculate Conditional Probability

The probability that income is at least \(100,000\) given that it is at least \(50,000\) is found by taking the higher income brackets \(100-499\) and \(\geq 500\), 0.100 and 0.006 respectively, and dividing by the probability of at least \(50,000\).\[ P(\text{at least } 100,000 \mid \text{at least } 50,000) = \frac{0.100 + 0.006}{0.321} \] \[ = \frac{0.106}{0.321} \approx 0.330 \]
05

Interpret the Results

The probability that a randomly chosen return shows an adjusted gross income of $50,000 or more is 0.321. The conditional probability that the income is at least $100,000, given that it is at least $50,000, is 0.330.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conditional Probability
Conditional probability is a fundamental concept that helps us find the probability of an event occurring given that another event has already happened.
It's like narrowing down the possible scenarios based on some known conditions. In this exercise, we wanted to find the probability that a tax return shows an income of at least \(100,000, given that it is known the return shows \)50,000 or more.
  • To calculate a conditional probability, you consider only the scenarios where the condition holds true (income 鈮� \(50,000 in our case).
  • Then, find how many of those scenarios also satisfy the additional condition (income 鈮� \)100,000).
The formula for conditional probability is given by: \[ P(B \mid A) = \frac{P(B \cap A)}{P(A)} \] For this exercise:
  • The probability of income being at least \(100,000 (event B) given that it is at least \)50,000 (event A) is: \[ P(\text{at least } 100,000 \mid \text{at least } 50,000) = \frac{0.106}{0.321} \approx 0.330 \]
This means that under the condition that the income is at least \(50,000, there is a 33% chance the income is \)100,000 or more.
Income Distribution
Income distribution refers to the way income is spread across a population. Understanding income distribution helps policymakers, economists, and researchers make informed decisions and analyses. This exercise focuses on adjusted gross income distribution from tax returns.
Income distribution is essential for:
  • Identifying economic inequality among different income brackets.
  • Developing tax policies that are fair and just.
  • Understanding the financial health of a population.
In this particular scenario, the income distribution shows different probabilities for income ranges like less than $25,000, $25,000-$49,000, $50,000-$99,000, $100,000-$499,000, and $500,000 or more.
The probabilities in the table enable us to compute further insights, such as the probability of having an income of $50,000 or more.
Adjusted Gross Income
Adjusted Gross Income (AGI) is a measure of income used to determine how much of your income is taxable. It plays a crucial role in tax computations and various income assessments.
Understanding AGI involves:
  • You start with your total income from all sources and then subtract specific deductions, like student loan interest or tuition fees, to get AGI.
  • AGI determines your eligibility for numerous deductions or credits available, helping in reducing your overall taxable income.
  • Being a pivotal figure in tax calculations, AGI can be found in Line 11 of Form 1040 in the U.S. tax return.
The significance of AGI is reflected in the fact that it sets the foundation for adjusted calculations and decisions on tax obligations. In this exercise, AGI distribution gives us the groundwork to calculate different probabilities and analyze income brackets and their probabilities on tax returns.

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Most popular questions from this chapter

Cold weather coming A TV weather man, predicting a colder-than-normal winter, said, "First, in looking at the past few winters, there has been a lack of really cold weather. Even though we are not supposed to use the law of averages, we are due." Do you think that "due by the law of averages" makes sense in talking about the weather? Why or why not?

Probability models? In each of the following situations, state whether or not the given assignment of probabilities to individual outcomes is legitimate, that is, satisfies the rules of probability. If not, give specific reasons for your answer. (a) Roll a 6-sided die and record the count of spots on the up-face: \(P(1)=0, P(2)=1 / 6, P(3)=1 / 3, P(4)=$$$1 / 3, P(5)=1 / 6, P(6)=0$$(b) Choose a college student at random and record gender and enrollment status: \)P(\( female full-time \))=0.56\(, \)P(\( male full-time \))=0.44, P(\( female part-time \))=0.24\( \)P(\( male part-time \))=0.17\( (c) Deal a card from a shuffled deck: \)P(\( clubs \))=12 / 52\(, \)P(\( diamond \)s)=12 / 52, P(\( heart \))=12 / 52, P(\( spades \))=\( \)16 / 52$

Nickels falling over You may feel it's obvious that the probability of a head in tossing a coin is about \(1 / 2\) because the coin has two faces. Such opinions are not always correct. Stand a nickel on edge on a hard, flat surface. Pound the surface with your hand so that the nickel falls over. Do this 25 times, and record the results. (a) What's your estimate for the probability that the coin falls heads up? Why? (b) Explain how you could get an even better estimate.

Monty Hall problem In Parade magazine, a reader posed the following question to Marilyn vos Savant and the "Ask Marilyn" column: Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say \(\\# 1,\) and the host, who knows what's behind the doors, opens another door, say # \(3,\) which has a goat. He says to you, "Do you want to pick door # \(2 ?\) Is it to your advantage to switch your choice of doors? \(^{4}\) The game show in question was Let's Make a Deal and the host was Monty Hall. Here's the first part of Marilyn's response: "Yes; you should switch. The first door has a \(1 / 3\) chance of winning, but the second door has a \(2 / 3\) chance." Thousands of readers wrote to Marilyn to disagree with her answer. But she held her ground. (a) Use an online Let's Make a Deal applet to perform at least 50 repetitions of the simulation. Record whether you stay or switch (try to do each about half the time) and the outcome of each repetition. (b) Do you agree with Marilyn or her readers? Explain.

Tossing coins Imagine tossing a fair coin 3 times. (a) What is the sample space for this chance process? (b) What is the assignment of probabilities to outcomes in this sample space?
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