Chapter 5: Q 66. (page 332)

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:
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. Probability shows an adjusted gross income of $$ 50, 000$ or more is $0.321$

b. Required conditional probability is $0.3302$

Step by step solution

Given Information

It is given that

Determining Probability for the randomly chosen return shows an adjusted gross income of $ 50,000 or more.

As per addition rule

$P (AorB) = P (A) + P (B)$

From table, probability of adjusted gross income of $50 - 99 (in thousands of $)$

$P (50 - 99) = 0.215$

Similarly $P (100 - 499) = 0.100$

$P (\geq 500) = 0.006$

Now, $P (income > $ 50, 000) = P (> 50)$

$= P (50 - 99) + P (100 - 499) + P (\geq 500)$

$= 0.215 + 0.100 + 0.006$

$= 0.321$

Hence, Probability shows an adjusted gross income of $$ 50, 000$ or more is $0.321$

Conditional Probability for income is at least $100,000

We know that $P (A ∣ B) = \frac{P (A and B)}{P (B)}$

From above part: $P (income > $ 50, 000) = 0.321$

As return shows income of atleast $$ 50, 000$

$\Rightarrow P (\geq $ 50, 000 and \geq $ 100, 000) = P (\geq $ 100, 000)$

$= P (100 - 499) + P (\geq 500)$

$= 0.100 + 0.006$

$= 0.106$

Using Conditional Probability

$P (\geq $ 100, 000 ∣ \geq $ 50, 000) = \frac{P (\geq $ 50, 000 and \geq $ 1, 00, 000)}{P (\geq $ 50, 000)}$

$= \frac{0.106}{0.321}$

$\approx 0.3302$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Given Information

Determining Probability for the randomly chosen return shows an adjusted gross income of $ 50,000 or more.

Conditional Probability for income is at least $100,000

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Pure Maths

Discrete Mathematics

Decision Maths

Geometry

Study anywhere. Anytime. Across all devices.