Student Debt. The Association of American Universities published a report titled "Looking More closely at student debt" This report explore the issue about the cost of college education and its impact on student loan debt. Using information from a credit reporting company, the following table provides a percentage distribution for the loan balance of outstanding student loans from individuals with graduate, professional, and under graduate degree debt.

Suppose that one of these individuals is selected at random.

Part (a) Without using the general addition rule, determine the probability that the individual obtained has a loan balance either between \(10,001 and \)100,000 inclusive, or at most $75,000.

Part (b) Obtain the probability in part (a) by using the general addition rule.

Part (c) Which method did you find easier?

Step by step solution

01

Part (a) Step 1. Given information.

The percentage distribution of outstanding student loan balances from individuals with graduate, professional, and undergraduate degree debt is shown in the table below:

Loan balance	Percentage
$1 -$10,000	43.1
$10,001 - $25,000	29.2
$ 25,001 -$50,000	16.5
$50,001 -$75,000	5.8
$75,001 -100,000	2.3
$100,001 or more	3.1

02

Part (a) Step 2. To determine the likelihood that the person you've just met has a loan balance of between $10,000 and $100,000, inclusive, or at most $75,000

If E is an event with a loan balance of $0 to $100,000, then the overall probability is:

$$\begin{gathered} P\left(E\right)=43.1+29.2+16.5+5.8+2.3+3.1 \\ P\left(E\right)=0.969 \end{gathered}$$

03

Part (b) Step 1. Using the general addition rule, find the probability in component (a).

Let $E_1$be the case where the individual's loan balance is between $10,000 and $100,000, and $E_2$is the case where the individual's loan balance is no more than $75,000.

$$\begin{gathered} P\left(E_1\right)=0.292+0.165+0.058+0.023=0.538 \\ P\left(E_2\right)=0.431+0.292+0.165+0.058=0.946 \\ P(E_1\cap E_2)=0.515 \end{gathered}$$

Now, apply the following general addition rule:

$$\begin{gathered} P(E_{1}or{E}_2)=P\left(E_1\right)+P\left(E_2\right)-P(E_1\cap E_2) \\ P(E_{1}or{E}_2)=0.538+0.946-0.515 \\ P(E_{1}or{E}_2)=0.969 \end{gathered}$$

04

Part (c) Step 1. To determine which way is the most convenient.

The first method is simpler since it requires less computation.

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