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Chapter 5: Problem 47

Education among young adults Choose a young adult (aged 25 to 29 ) at random. The probability is 0.13 that the person chosen did not complete high school, 0.29 that the person has a high school diploma but no further education, and 0.30 that the person has at least a bachelor's degree. (a) What must be the probability that a randomly chosen young adult has some education beyond high school but does not have a bachelor's degree? Why? (b) What is the probability that a randomly chosen young adult has at least a high school education? Which rule of probability did you use to find the answer?

Short Answer

Expert verified
(a) 0.28 (b) 0.87, using the rule of probability addition for mutually exclusive events.

Step by step solution

01

Define the Event Probabilities

We are given several probabilities regarding the education levels of young adults aged 25 to 29. The probabilities are as follows: \( P(\text{Did not complete high school}) = 0.13 \), \( P(\text{Has a high school diploma only}) = 0.29 \), and \( P(\text{Has at least a bachelor's degree}) = 0.30 \). The sum of these probabilities does not include young adults who have some education beyond high school but do not have a bachelor's degree.
02

Calculate Probability for Some Education Beyond High School

To find the probability of a young adult having some education beyond high school but not completing a bachelor's degree, we start with the total probability, which is 1. Subtract the probabilities of all other categories (those not in this category): Thus, this probability is given by \( 1 - P(\text{Did not complete high school}) - P(\text{Has a high school diploma only}) - P(\text{Has at least a bachelor's degree}) \). That is \( 1 - 0.13 - 0.29 - 0.30 = 0.28 \).
03

Calculate Probability for At Least High School Education

To find the probability that a young adult has at least a high school education, sum the probabilities of having only a high school diploma or having more than a high school diploma. This is calculated as \( P(\text{Has a high school diploma only}) + P(\text{Some college but not bachelor's}) + P(\text{Has a bachelor's degree }) \). Since we found that \( P(\text{Some college but not bachelor's}) = 0.28 \) in Step 2, we add all three: \( 0.29 + 0.28 + 0.30 = 0.87 \). This uses the rule of simple addition of probabilities for mutually exclusive events.

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Key Concepts

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

Educational Attainment Statistics
Educational Attainment Statistics play a crucial role in understanding the overall educational levels within a specific age group or population. Such statistics provide insights into how many individuals have reached certain levels of education, such as high school diplomas, bachelor's degrees, or other certifications. By analyzing these statistics, educators and policymakers can identify trends, promote higher education, and address educational gaps.
  • They help in measuring the educational outcomes of a population.
  • They highlight the proportion of individuals at different educational stages.
  • These statistics can guide the allocation of resources to different educational programs.
Understanding these statistics for young adults aged 25 to 29 can inform future educational policies. For instance, if a significant portion of this group hasn't completed high school, targeted interventions can be developed to encourage further education.
Probability Rules
Probability rules are fundamental mathematical principles that determine the likelihood of an event occurring. In the context of educational attainment statistics, probability helps in calculating the chances of a young adult falling into a particular educational category. There are several rules of probability that are vital:
  • The sum of probabilities for all possible outcomes of a random experiment is equal to 1.
  • The probability of an event is calculated as a fraction, where the numerator is the favorable outcomes and the denominator is the total number of possible outcomes.
  • Probabilities range between 0 and 1, inclusive.
In our exercise, the rule of complementary probabilities was used. This rule states that the probability of an event not occurring is equal to 1 minus the probability of the event occurring. So, by subtracting given probabilities of specific educational levels from 1, we find the probability of other educational levels.
Mutually Exclusive Events
Mutually exclusive events are scenarios where the occurrence of one event means the other cannot happen simultaneously. In probability, understanding these events helps in calculating accurate probabilities. For the educational categories of the exercise, each is mutually exclusive:
  • A young adult can either not complete high school, have a high school diploma only, or have further education.
  • No two events can happen at the same time, e.g., one cannot both complete high school and not complete it at the same time.
When events are mutually exclusive, the probability of either event occurring is simply the sum of their individual probabilities. This principle was essential for part (b) of the exercise, where the total probability of having at least a high school education was calculated by adding these distinct probabilities.
High School Education Levels
High School Education Levels refer to the basic educational background someone attains typically by the age of 18. This stage is crucial, as it sets the foundation for further education, career paths, and personal development. Understanding the distribution of high school education levels among young adults is integral for educators and policy makers:
  • "Did not complete high school" indicates individuals who might require additional educational support.
  • "High school diploma only" reflects those who have completed basic education but might need motivation to pursue higher education.
  • People with education beyond high school, including bachelor's degrees, tend to have wider career opportunities.
Educational statistics for this level help understand how many young adults complete their high school education and how many move beyond it, aiding in resource allocation and development of educational strategies.

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