Problem 57 - 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 exactly one is elderly? (Round to the nearest tenth of a percent.)

Step-by-Step Solution

Answer

The percent chance is 36.9%.

Step 1: Calculate the Probability of a Citizen Being Elderly

First, determine the probability of a randomly selected citizen being elderly. Divide the number of elderly citizens by the total population: \( P(E) = \frac{40 \text{ million}}{317 \text{ million}} \approx 0.126 \).

Step 2: Calculate the Probability of a Citizen Not Being Elderly

Next, determine the probability of a citizen not being elderly. This is the complement of being elderly: \( P(N) = 1 - P(E) \approx 1 - 0.126 = 0.874 \).

Step 3: Use the Binomial Probability Formula

To find the probability that exactly one out of five citizens is elderly, use the binomial probability formula: \[P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\] where \(n = 5\), \(k = 1\), and \(p = 0.126\).

Step 4: Calculate the Binomial Coefficient

Calculate the binomial coefficient \(\binom{5}{1}\), which is the number of ways to choose 1 citizen to be elderly out of 5: \(\binom{5}{1} = 5\).

Step 5: Compute the Probability using the Binomial Formula

Substitute the values into the binomial formula: \[P(X=1) = 5 \times (0.126)^1 \times (0.874)^4 \] Calculate each part: \((0.126)^1 = 0.126\) and \((0.874)^4 \approx 0.586\). Multiply these together: \[P(X=1) = 5 \times 0.126 \times 0.586 \approx 0.369\]

Step 6: Convert Probability to Percent

Convert the probability to a percentage: \(0.369 \times 100 \approx 36.9\%\). Round to the nearest tenth of a percent.

Key Concepts

Probability of Elderly

To understand the probability of meeting an elderly person, we start by defining what we mean by 'elderly' in this context. An elderly person is someone who is aged 65 or older. In 2013, the proportion of citizens in the U.S. who were considered elderly was determined by dividing the number of elderly people by the total population. Here's how it's calculated:

• Total U.S. population: 317 million
• Number of elderly people: 40 million
• Probability of a citizen being elderly, \( P(E) \), is calculated as \( P(E) = \frac{40}{317} \approx 0.126 \).

This means, there's roughly a 12.6% chance that a single, randomly chosen U.S. citizen is elderly. This is the foundation step in solving the overall exercise problem.

Binomial Coefficient

The binomial coefficient is a mathematical concept used in probability when we deal with situations that have two possible outcomes, like being elderly or not. In this exercise, we use the binomial coefficient to calculate the probability of exactly one out of five citizens being elderly.
The binomial coefficient, denoted as \( \binom{n}{k} \), represents the number of ways to choose \( k \) successes from \( n \) trials. In our example:

• \( n = 5 \) (representing the five citizens you meet)
• \( k = 1 \) (representing exactly one being elderly)
• The binomial coefficient is calculated as \( \binom{5}{1} = 5 \).

This calculation shows us there are 5 different ways for exactly one out of five people to be elderly. The coefficient is a crucial part of using the binomial formula to find specific probabilities.

Percent Probability

After calculating the probability using the binomial formula, the next important step is converting this probability into a percentage. This is helpful because percentages can be easier to interpret and compare in real-life scenarios.
To convert a decimal probability to a percentage:

• Multiply the decimal by 100.
• Then, if necessary, round it to the desired precision.

For example, in this exercise, we found that the probability of exactly one out of five citizens being elderly is approximately 0.369. So, the conversion would be:\[ 0.369 \times 100 = 36.9\% \]This percent probability tells us that there is a 36.9% chance of meeting exactly one elderly person among five randomly selected U.S. citizens, giving a clearer understanding of likelihood in everyday terms.