Problem 57 - College Algebra with Corequisite Support

Question

Use this data for the exercises that follow: In \(2013,\) there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over). \(^{2}\) If you meet five U.S. citizens, what is the percent chance that exactly one is elderly? (Round to the nearest tenth of a percent.)

Step-by-Step Solution

Answer

The percent chance is approximately 36.9%.

Step 1: Understand the Problem

We need to find the probability that exactly one of the five randomly selected U.S. citizens is elderly, aged 65 or over.

Step 2: Determine Elderly Probability

First, we calculate the probability of finding an elderly person among the general population. It's calculated as the number of elderly people divided by the total population: \(\frac{40\text{ million}}{317\text{ million}} \approx 0.126\).

Step 3: Determine Non-Elderly Probability

Next, find the probability of a randomly selected person not being elderly, which is \(1 - 0.126 = 0.874\).

Step 4: Use Binomial Probability Formula

We use the binomial formula to find the probability of getting exactly one elderly citizen out of five: \[P(X=k) = \binom{n}{k} p^k (1-p)^{n-k},\] where \(n=5\), \(k=1\), and \(p=0.126\). Substituting these values gives: \[P(X=1) = \binom{5}{1} (0.126)^1 (0.874)^4\].

Step 5: Calculate the Binomial Coefficient

The binomial coefficient \(\binom{5}{1}\) is calculated as \(\frac{5!}{1!(5-1)!} = 5\).

Step 6: Calculate the Probability Components

Calculate \((0.126)^1 = 0.126\) and \((0.874)^4 \approx 0.586\).

Step 7: Complete the Binomial Probability Calculation

Substitute these values into the binomial formula: \[P(X=1) = 5 \cdot 0.126 \cdot 0.586\]. Compute this to get approximately \(0.369\).

Step 8: Convert Probability to Percent

Convert the probability \(0.369\) to a percentage by multiplying by 100, yielding \(36.9\%\). Round this to the nearest tenth of a percent.

Key Concepts

Elderly Population

The concept of the elderly population is crucial when discussing demographic statistics. In this context, 'elderly' typically refers to individuals aged 65 and over. For the year 2013 in the United States, there were approximately 40 million such individuals. Understanding this demographic is important for various reasons, including:

• Planning for social services and healthcare needs.
• Understanding economic impacts, such as retirement funding.
• Studying the effects of an aging population on workforce and productivity.

When calculating probabilities, it's vital to correctly identify and use the right population figures, as we see in this exercise where the proportion of elderly citizens plays a central role.

Probability Calculation

Probability calculation is a fundamental concept in statistics, allowing us to quantify how likely an event is to occur. In our scenario, we want to determine the chances of meeting exactly one elderly person out of five randomly chosen U.S. citizens. To start, we calculate the probability of encountering an elderly person within the general population. This is done by dividing the number of elderly by the total number of people in the population, which offers a probability of 0.126. Understanding this initial calculation sets the foundation for further analysis, where specific statistical methods are used to explore complex scenarios, like meeting a certain number of elderly individuals among a group. Probability is expressed as a value between 0 and 1, with 1 indicating a certainty.

Binomial Coefficient

The binomial coefficient is a key element in binomial probability calculations. It represents the number of ways you can choose a subset of elements from a larger set, and is notated as \(\binom{n}{k}\), where \(n\) is the total number of elements to choose from, and \(k\) is the number of picks.In our example, we use \(\binom{5}{1}\) to determine the probability of picking exactly one elderly person from a group of five. Calculating this, we find:\[\binom{5}{1} = \frac{5!}{1!(5-1)!} = 5\]The coefficient 5 tells us there are five possible combinations to achieve the desired outcome.This coefficient is used within the formula to not only count the possible combinations but also weigh the probability of each specific occurrence happening within those combinations.

Statistics in Education

Statistics play an increasingly critical role in education, helping educators and students alike make data-driven decisions. In exercises like the one given, students apply statistical concepts, such as probability and binomial coefficients, to realistic scenarios. Using statistics in education involves:

• Problem-solving skills—analyzing real-world data to derive insights.
• Application of mathematical theories to practical situations.
• Development of critical thinking by evaluating and interpreting data.

Exercises like these provide hands-on experience, preparing students for more complex decision-making processes in their careers. Encouraging students to explore these concepts deeply fosters a better understanding of critical statistical methodologies.