Problem 58 - College Algebra

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). \(^{[34]}\) If you meet five U.S. citizens, what is the percent chance that three are elderly? (Round to the nearest tenth of a percent.)

Step-by-Step Solution

Answer

The percent chance is approximately 1.5%.

Step 1: Identify the probabilities

First, identify the probability of meeting an elderly person. There were 40 million elderly people out of 317 million total citizens. Therefore, the probability \( P(E) \) that a person is elderly is \( P(E) = \frac{40}{317} \approx 0.126 \) or 12.6%.

Step 2: Set up the Binomial Distribution

We need to calculate the probability of exactly 3 out of 5 citizens being elderly using the binomial distribution. The binomial probability formula is \( P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( n \) is the number of trials (5 citizens), \( k \) is the number of successes (3 elderly citizens), and \( p \) is the probability of success (0.126).

Step 3: Calculate the Binomial Coefficient

The binomial coefficient \( \binom{5}{3} \) is the number of ways to choose 3 successes (elderly citizens) out of 5 trials:\[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

Step 4: Apply the Binomial Formula

Use the binomial formula to find \( P(X=3) \):\[ P(X=3) = \binom{5}{3} (0.126)^3 (1-0.126)^{5-3} \]\[ P(X=3) = 10 \cdot (0.126)^3 \cdot (0.874)^2 \]

Step 5: Perform the Calculations

Calculate the values:\( (0.126)^3 = 0.002 \) and \( (0.874)^2 = 0.764 \).Thus, \( P(X=3) = 10 \times 0.002 \times 0.764 = 0.015 \).

Step 6: Convert Probability to Percent

Convert the probability to a percentage by multiplying by 100:\[ 0.015 \times 100 = 1.5 \%\]Thus, the percent chance that exactly 3 out of 5 U.S. citizens are elderly is approximately 1.5%.

Key Concepts

Probability

Understanding probability is crucial when determining the likelihood of events in various scenarios, such as the chance of meeting elderly people among U.S. citizens. Probability measures how likely an event is to happen.

A simple probability can be calculated by dividing the number of favorable outcomes by the total number of outcomes. In the exercise, the probability of meeting an elderly person, denoted as \( P(E) \), is calculated by dividing the number of elderly citizens (40 million) by the total number of citizens (317 million). This gives us a probability of 0.126, or 12.6%.

Probabilities are typically expressed as fractions, decimals, or percentages. It's important to remember probabilities range from 0 to 1, with 0 meaning impossible, 1 meaning certain, and any value in between representing the likelihood of the event occurring. Converting probability to a percentage can help visualize this likelihood on a common scale.

Binomial Coefficient

The binomial coefficient is a key part of calculating probabilities in a binomial distribution. It represents the number of ways to pick a specific number of successful outcomes from a total number of trials.

In mathematical notation, it is written as \( \binom{n}{k} \), where \( n \) is the total number of trials, and \( k \) is the number of successful outcomes we're interested in. This is also known as "n choose k."

The formula for calculating the binomial coefficient is:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

For example, in the exercise, we calculated \( \binom{5}{3} \) for choosing 3 elderly citizens out of 5. This can be computed as \( \frac{5!}{3!(5-3)!} \), ultimately resulting in 10 different ways to pick these 3 elderly citizens from the group.

Statistics

Statistics is the study of data: how to collect, summarize, and analyze it to understand and use it effectively. The exercise utilizes statistical methods to understand the likelihood of encountering elderly individuals among randomly encountered U.S. citizens.

This problem employs the concept of a binomial distribution, a statistical model that describes the outcomes of experiments where there are fixed numbers of trials, two possible outcomes per trial (success or failure), and a consistent probability of success.

The binomial distribution gives us a much-needed framework to work with, explaining the distribution of probabilities across different outcomes. It helps answer questions about natural or random processes where only two outcomes occur, like determining if individuals fit a certain demographic.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It forms the foundation for setting up and solving the equations required in the exercise.

Working through this problem involves using algebraic concepts to manipulate and solve the probability formula for binomial distribution:

• \( P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \)

This equation gives the probability of \( k \) successes (elderly people, in this case) in \( n \) trials (citizens encountered), with \( p \) being the probability of success.

Maintaining accuracy in the manipulation and calculation of these equations is key to determining the correct likelihood or probability in various situations, demonstrating the power and necessity of algebra in statistical computations.