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Chapter 7: Problem 27

A study conducted by the Metro Housing Agency in a midwestern city revealed the following information concerning the age distribution of renters within the city. $$\begin{array}{lcc} \hline & & \text { Group } \\ \text { Age } & \text { Adult Population, \% } & \text { Who Are Renters, \% } \\\ \hline 21-44 & 51 & 58 \\ \hline 45-64 & 31 & 45 \\ \hline 65 \text { and over } & 18 & 60 \\ \hline \end{array}$$ a. What is the probability that an adult selected at random from this population is a renter? b. If a renter is selected at random, what is the probability that he or she is in the \(21-44\) age bracket? c. If a renter is selected at random, what is the probability that he or she is 45 yr of age or older?

Short Answer

Expert verified
a. The probability that an adult selected at random from this population is a renter is \(P(renter) = \frac{51 * 58 + 31 * 45 + 18 * 60}{100} \). b. If a renter is selected at random, the probability that he or she is in the $21-44$ age bracket is \(P(21-44 | renter) = \frac{(51 * 58) / 100}{P(renter)}\). c. If a renter is selected at random, the probability that he or she is 45 years of age or older is \(P(45 \text{ or older} | renter) = \frac{[(31 * 45) + (18 * 60)] / 100}{P(renter)}\).

Step by step solution

01

a. Probability of an adult being a renter

First, we need to calculate the total number of adults who are renters. We do this by multiplying the percentage of the adult population in each age group with the percentage of renters in that group. Then, we sum the results for each age group and compute the overall probability. Probability of being a renter = (Percentage of adult population in each age group * Percentage of renters in each age group) / 100 \(P(renter) = [51 * 58 + 31 * 45 + 18 * 60] / 100\)
02

b. Probability of a random renter being 21-44 yrs old

Here, we need to find the probability of selecting a renter who is between 21 and 44 years old. We will use the conditional probability concept to solve this: \(P(21-44 | renter) = \frac{P(renter \cap 21-44)}{P(renter)}\) We already calculated \(P(renter)\) in part a. Now, we need to calculate \(P(renter \cap 21-44)\), which can be found by multiplying the percentages of adult population and renters in the 21-44 age group: \(P(renter \cap 21-44) = (51 * 58) / 100\)
03

c. Probability of a random renter being 45 yrs or older

For this part, we need to find the probability of selecting a renter who is 45 years old or older. Similar to part b, we will use the conditional probability concept: \(P(45 \text{ or older} | renter) = \frac{P(renter \cap 45 \text{ or older})}{P(renter)}\) Again, we know the value of \(P(renter)\) from part a. We can compute \(P(renter \cap 45 \text{ or older})\) by summing the probabilities of renters in the 45-64 age group and the 65 and over age group: \(P(renter \cap 45 \text{ or older}) = [(31 * 45) + (18 * 60)] / 100\)

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Key Concepts

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

Age Distribution
Age distribution is a key factor in understanding the structure of a population. It provides insights into the different age groups within a population and their proportions relative to the whole. In this example, the population of a city has been divided into three age groups: 21-44 years, 45-64 years, and 65 years and older. Each group makes up a certain percentage of the adult population. This breakdown allows researchers to see how the population is spread across different age categories.
For instance:
  • The 21-44 age group constitutes 51% of the adult population,
  • The 45-64 group accounts for 31%, and
  • The 65 and older group comprises 18%.
By examining this distribution, we can gain a better understanding of various societal dynamics, such as housing needs, workforce composition, and healthcare services requirements.
Conditional Probability
Conditional probability is a measure of the likelihood of an event occurring, given that another event has already occurred. It is denoted as \(P(A | B)\), which reads as the probability of A occurring given B has occurred. This concept is essential when determining probabilities in dependent scenarios.
In the context of the exercise, if a renter is selected at random, conditional probability helps us find the likelihood that they belong to a certain age group, such as 21-44 years. To calculate this, we need to know the joint probability, which is the probability of both being a renter and belonging to the 21-44 age group, divided by the general probability of being a renter.
To illustrate:
  • First, find the joint probability of a renter being 21-44, calculated by multiplying the percentages relevant to the age group.
  • Next, use the previously calculated probability of a general renter \(P(renter)\) to find \(P(21-44 | renter)\).
This method allows for precise calculations of probabilities in specific contexts, helping to inform decisions based on available data.
Percentage Calculations
Percentage calculations are a straightforward method to represent numbers as parts of a hundred, making comparisons and analyses more intuitive. They are used to quantify parts of a whole in a clear manner. In probability and statistics, percentages are often converted back to decimals or used to calculate likelihoods and comparisons.
Consider the step to find the probability of being a renter. We use the percentage of the adult population that belongs to each age group and multiply it with the respective renting percentages. This involves straightforward multiplication and addition operations:
  • Multiply each group's adult population percentage by their renter's percentage.
  • Add these results together to get a total percentage value.
Finally, convert this percentage to a decimal to express it as a probability. Using percentages simplifies these tasks and makes the results easy to interpret in practical terms.
Probability Distribution
A probability distribution describes how the probabilities of different outcomes are distributed across a sample space. In the context of age and renting status, each age group has specific probabilities associated with being renters.
In this scenario, we analyze the probability distribution across different age brackets:
  • The 21-44 age group has a 58% probability of renting,
  • The 45-64 group records a 45% probability, and
  • The 65 and older group shows a 60% probability.
Understanding these distributions is key to comprehending the overall renting dynamics within the population. This can help housing agencies and policymakers provide better accommodations or support services.
This concept of probability distribution enables detailed analysis of not just general renting trends but also tailored strategies to handle specific segments contingent upon age and other demographic factors.

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