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Chapter 5: Problem 10

The probability that a randomly selected individual in the United States 25 years and older has at least a bachelor's degree is \(0.094 .\) The probability that an individual in the United States 25 years and older has at least a bachelor's degree, given that the individual lives in Washington \(\mathrm{DC},\) is, \(0.241 .\) Are the events "bachelor's degree" and "lives in Washington, DC," independent?

Short Answer

Expert verified
The events 'bachelor's degree' and 'lives in Washington, DC' are not independent.

Step by step solution

01

- Identify Given Probabilities

Let A be the event that an individual has at least a bachelor's degree, and B be the event that the individual lives in Washington, DC. We are provided with the following probabilities: 1. The probability that an individual has at least a bachelor's degree, P(A) = 0.094.2. The probability that an individual has at least a bachelor's degree given that they live in Washington, DC, P(A|B) = 0.241.
02

- Define Independent Events

Two events, A and B, are independent if and only if the occurrence of one does not affect the probability of the occurrence of the other. Mathematically, this can be expressed as: P(A|B) = P(A).
03

- Check for Independence

Compare the given probabilities to test for independence.From the problem, we know:P(A) = 0.094P(A|B) = 0.241If the events were independent, P(A|B) should be equal to P(A). Clearly, P(A|B) = 0.241 is not equal to P(A) = 0.094.
04

- Conclude

Since P(A|B) is not equal to P(A), the events 'bachelor's degree' and 'lives in Washington, DC' are not independent.

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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 the probability of an event occurring given that another event has already occurred. It's represented as \( P(A|B) \), which reads as 'the probability of A given B'. In our exercise, \( P(A|B) = 0.241 \), meaning there is a 24.1% chance an individual has at least a bachelor's degree if they live in Washington, DC. This concept helps us understand how probabilities change when we have additional information. It's fundamental in fields like statistics and machine learning.
Probability Theory
Probability theory is the branch of mathematics that deals with calculating the likelihood of a given event's occurrence, expressed as a number between 0 and 1. In the context of our exercise, we encountered basic probability terms and ideas. We used the following:
  • \( P(A) \): Probability that an individual has at least a bachelor's degree (\(0.094\))
  • \( P(A|B) \): Probability that an individual has at least a bachelor's degree, given that they live in Washington, DC (\(0.241\))

Probability theory provides the tools to evaluate and understand such probabilities, enabling us to draw conclusions and make decisions based on data.
Statistical Independence
Statistical independence describes the relationship between two events where the occurrence of one does not affect the occurrence of the other. For two events, A and B, to be independent, the conditional probability \( P(A|B) \) must be equal to the marginal probability \( P(A) \). In our exercise:
  • \( P(A) = 0.094 \)
  • \( P(A|B) = 0.241 \)

Since \( P(A|B) \) is not equal to \( P(A) \), the events 'bachelor's degree' and 'lives in Washington, DC' are not independent. Understanding independence helps in simplifying complex probability problems.
Education Level Statistics
Education level statistics analyze data related to educational attainment, such as the percentage of individuals with bachelor's degrees. This field provides insights into how education may vary based on different factors like geography, age, or socioeconomic status. In this exercise, we see:
  • The probability that any U.S. individual, 25 and older, has at least a bachelor's degree is \(0.094\)
  • The probability that an individual in Washington, DC, 25 and older, has at least a bachelor's degree is significantly higher at \(0.241\)

These statistics highlight disparities in educational attainment between different regions. By analyzing such data, policymakers and educators can tailor efforts to improve education equity and access.

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Most popular questions from this chapter

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