Problem 27 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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?

Step-by-Step Solution

Answer

a. The probability that an adult selected at random is a renter is approximately 0.5433. b. The probability that a renter is in the 21-44 age bracket is approximately 0.544. c. The probability that a renter is 45 years of age or older is approximately 0.455.

Step 1: Calculate the probability of each age group being a renter

Let's calculate the probability of each age group being a renter using the given percentages. For each age group: \(P(R \mid A) = \frac{\% \text { Who Are Renters }}{100}\). For the 21-44 age group: \(P(R \mid A_{21-44}) = \frac{58}{100} = 0.58\) For the 45-64 age group: \(P(R \mid A_{45-64}) = \frac{45}{100} = 0.45\) For the 65 and over age group: \(P(R \mid A_{65+}) = \frac{60}{100} = 0.60\)

Step 2: Determine the probabilities of each age group

To simplify calculations, let's convert the given percentages of adult population to probabilities. For the 21-44 age group: \(P(A_{21-44}) = \frac{51}{100} = 0.51\) For the 45-64 age group: \(P(A_{45-64}) = \frac{31}{100} = 0.31\) For the 65 and over age group: \(P(A_{65+}) = \frac{18}{100} = 0.18\)

Step 3a: Calculate the probability of an adult being a renter

To determine the overall probability of an adult being a renter (\(P(R)\)), we will use the law of total probability: \(P(R) = P(R \mid A_{21-44})P(A_{21-44})+P(R \mid A_{45-64})P(A_{45-64})+P(R \mid A_{65+})P(A_{65+})\). \(P(R) = 0.58 \times 0.51 + 0.45 \times 0.31 + 0.60 \times 0.18 = 0.2958 + 0.1395 + 0.108 = 0.5433\) The probability that an adult selected at random is a renter is approximately 0.5433.

Step 3b: Calculate the probability that a renter is in the 21-44 age bracket

Now, we want to calculate the probability that a renter is in the 21-44 age bracket, given that they are a renter (\(P(A_{21-44} \mid R)\)). Using the formula for conditional probability: \(P(A_{21-44} \mid R) = \frac{P(R \cap A_{21-44})}{P(R)} = \frac{P(R \mid A_{21-44})P(A_{21-44})}{P(R)}\) \(P(A_{21-44} \mid R) = \frac{0.58 \times 0.51}{0.5433} = \frac{0.2958}{0.5433} \approx 0.544\) The probability that a renter is in the 21-44 age bracket is approximately 0.544.

Step 3c: Calculate the probability that a renter is 45 years of age or older

Finally, we want to calculate the probability that a renter is 45 years of age or older, given that they are a renter (\(P(A_{45+} \mid R)\)). Since there are only two age groups for 45 years or older, we can sum their probabilities: \(P(A_{45+} \mid R) = P(A_{45-64} \mid R) + P(A_{65+} \mid R)\). Using the formula for conditional probability: \(P(A_{45-64} \mid R) = \frac{P(R \cap A_{45-64})}{P(R)} = \frac{P(R \mid A_{45-64})P(A_{45-64})}{P(R)}\) and \(P(A_{65+} \mid R) = \frac{P(R \cap A_{65+})}{P(R)} = \frac{P(R \mid A_{65+})P(A_{65+})}{P(R)}\). \(P(A_{45+} \mid R) = \frac{0.45 \times 0.31}{0.5433} + \frac{0.60 \times 0.18}{0.5433} = \frac{0.1395}{0.5433} + \frac{0.108}{0.5433} \approx 0.256 + 0.199 \approx 0.455\) The probability that a renter is 45 years of age or older is approximately 0.455.

Key Concepts

Age Distribution

Age distribution is a vital statistic in social studies. It helps understand how different age groups interact with society, such as in renting habits within a city. For this exercise, we are looking at three age groups: 21-44 years, 45-64 years, and 65 years and over. Understanding how population is divided across these age groups is critical to predicting behaviors.

• The 21-44 age group represents 51% of the adult population.
• The 45-64 age group makes up 31%.
• The 65 and over age group constitutes 18%.

Each group's rental habits differ, which provides insight into the preferences and economic circumstances of these age groups. The intersection of these distribution percentages helps in identifying the likelihood of individuals within a specific age bracket being renters. It's also the first step in calculating probabilities related to this population.

Conditional Probability

Conditional probability helps us calculate the likelihood of an event, based on prior knowledge of related occurrences. In this context, we are interested in probabilities such as the likelihood that a randomly picked renter belongs to a specific age group.For example, to find the probability that a renter falls in the 21-44 age group, given they are indeed renters, we use:\[ P(A_{21-44} \mid R) = \frac{P(R \cap A_{21-44})}{P(R)} \]This formula shows how we take the product of probabilities of the renter being in the 21-44 age group and divide it by the overall probability of being a renter. The conditional probability offers a nuanced view by focusing on subsections of a population affected by specific conditions like age or rentership.Ultimately, this allows us to understand not just the raw data of renters' ages, but how age and rental status are related.

Law of Total Probability

The Law of Total Probability is crucial for calculating broad probabilities when multiple independent events are connected. This law helps compute the overall probability of an adult being a renter, by taking into account all age groups.The law states:\[ P(R) = P(R \mid A_{21-44})P(A_{21-44}) + P(R \mid A_{45-64})P(A_{45-64}) + P(R \mid A_{65+})P(A_{65+}) \]This equation sums the probabilities of being a renter within each age group, weighted by the population proportion of each group. By breaking the entire population into distinct subgroups, it provides a comprehensive overview. Using this law, we established the probability of an adult renter at approximately 0.5433. Thus, integrating these individual probabilities affords us a complete picture of the rental landscape based on age distributions.