91Ó°ÊÓ

.Student Debt. At the end of 2016 , the average outstanding student debt for bachelor's degree recipients was \(\$ 28,500\). Here is the distribution of outstanding education debt (in thousands of dollars): 17 \begin{tabular}{|l|c|c|c|c|c|c|} \hline Debt & \(<10\) & 10 to \(<20\) & 20 to \(<30\) & 30 to \(<40\) & 40 to \(<50\) & \(\geq 50\) \\ \hline Probability & \(0.40\) & \(0.13\) & \(0.17\) & \(0.12\) & \(0.08\) & \(0.10\) \\ \hline \end{tabular} a. What is the probability that a randomly chosen student has an outstanding debt of \(\$ 20,000\) or more? b. Given that a student has an outstanding debt of at least \(\$ 20,000\), what is the conditional probability that the debt is at least \(\$ 50,000 ?\)

Step by step solution

01

Understand the Problem

We are given a probability distribution for student debts. Our task is to first determine the probability that a student's debt is at least $20,000. Then, we find the conditional probability that given a student owes at least $20,000, the debt is $50,000 or more.

02

Calculate Total Probability for Debt ≥ $20,000

To find the probability that a student owes \(20,000 or more, sum the probabilities of the corresponding ranges from the table. This includes the ranges \)20 \ to <30\(, \)30 \ to <40\(, \)40 \ to <50\(, and \)\geq 50$. \[ P(\text{debt} \geq 20000) = 0.17 + 0.12 + 0.08 + 0.10 = 0.47 \]

03

Find Probability for Debt ≥ $50,000

From the table, directly read the probability that a student has a debt of at least $50,000. \[ P(\text{debt} \geq 50000) = 0.10 \]

04

Calculate Conditional Probability

We need the conditional probability that a student's debt is at least \(50,000 given that their debt is at least \)20,000. Use the formula for conditional probability: \[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \] where event A is "debt ≥ \(50,000" and event B is "debt ≥ \)20,000".\[ P(\text{debt} \geq 50000 \mid \text{debt} \geq 20000) = \frac{P(\text{debt} \geq 50000)}{P(\text{debt} \geq 20000)} = \frac{0.10}{0.47} \approx 0.213 \]

05

Finalize Solution

We have now calculated that the probability a student has a debt of $20,000 or more is 0.47. Additionally, given that a student has a debt of $20,000 or more, the probability that the debt is at least $50,000 is approximately 0.213.

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Key Concepts

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

Student Debt

Student debt is a substantial issue for many college graduates. In 2016, the average debt for bachelor's degree holders was around $28,500. Understanding how this debt is distributed among students is crucial for grasping its impact.

The distribution of student debt can be described through a probability distribution, which shows the likelihood of various debt amounts.

• A large portion, 40%, owes less than $10,000.
• Other ranges include debts between $10,000 to $20,000 (13%) and debts between $20,000 to $30,000 (17%).
• Larger debts, such as those between $40,000 to $50,000, are less common, representing a probability of 8%.
• Finally, 10% owe $50,000 or more.

This data helps us understand the probability of student debt and forms the basis for exploring conditional probability.

Conditional Probability

Conditional probability is key for understanding the chance of one event given another event has already occurred. It answers questions like: "What is the likelihood of event A, given event B?"

In the context of our problem, we are interested in knowing the probability that a student's debt is at least \(50,000, given that we already know their debt is at least \)20,000. The formula to calculate this is:

\[ P(A \,|\, B) = \frac{P(A \cap B)}{P(B)} \]

Here:

• Event A = "Debt is at least \(50,000"
• Event B = "Debt is at least \)20,000"

Using our exercise data, we find:

• \(P(A \cap B)\) is the probability of debt being at least \(50,000, which is 0.10.
• \(P(B)\) is the probability of debt being at least \)20,000, calculated by summing the probabilities of all relevant categories from $20,000 upwards.
• Thus, \(P(B) = 0.17 + 0.12 + 0.08 + 0.10 = 0.47\)

Finally, the conditional probability is approximately 0.213, indicating the likelihood of high debt among those with significant student loans.

Probability Calculation

Probability calculation allows us to quantify uncertainty. In situations like calculating student debt probabilities, it's about determining the likelihood of different amounts of debt.

To solve such probability problems, follow these simple steps:

• Identify events and their probabilities from the data provided.
• For total probabilities, sum the probabilities of all relevant events.
• For conditional probabilities, use the formula for conditional probability to find the likelihood of one event occurring under the condition of another.

For instance, to find the probability that a student debt is $20,000 or more, consider all relevant categories:
\[ P( ext{debt} \geq 20000) = 0.17 + 0.12 + 0.08 + 0.10 = 0.47 \]
For conditional probability, use:
\[ P(A \mid B) = \frac{P(A)}{P(B)} \quad \text{using} \quad P(A)=0.10 \quad \text{and} \quad P(B)=0.47. \]
This method helps break down complex problems into simpler components, making probability concepts easier to grasp.