Problem 59 - 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 four are elderly? (Round to the nearest thousandth of a percent.)

Step-by-Step Solution

Answer

The chance is approximately 0.090%.

Step 1: Determine Probability of a Single Individual Being Elderly

First, calculate the probability that a randomly selected U.S. citizen is elderly. There are 40 million elderly individuals out of a total of 317 million citizens. So, the probability \( P(E) \) of selecting an elderly person is given by \[ P(E) = \frac{40}{317}. \] By calculating, \( P(E) \approx 0.126 \).

Step 2: Calculate Probability of Four Out of Five Being Elderly

Use the binomial probability formula \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( n \) is the number of citizens (5), \( k \) is the number we want to be elderly (4), and \( p \) is the probability of selecting an elderly person (0.126).\[ P(X = 4) = \binom{5}{4} (0.126)^4 (1-0.126)^{5-4}. \] Compute this as \( \binom{5}{4} = 5 \), so \( P(X = 4) = 5 \times (0.126)^4 \times (0.874)^1 \approx 0.000899 \).

Step 3: Convert Probability to Percentage

Convert the probability from Step 2 to a percentage by multiplying by 100 and then rounding to the nearest thousandth of a percent. \[ 0.000899 \times 100 \approx 0.0899\% \] Rounded, this is approximately \( 0.090\% \).

Key Concepts

Binomial distribution

The binomial distribution is a fundamental concept in probability theory and statistics. It allows us to model situations where there are two possible outcomes (success or failure) for each trial and the trials are independent. In the context of our exercise, we use the binomial distribution to calculate the likelihood of meeting a certain number of elderly people out of five U.S. citizens. The key parameters here are:

• **The number of trials** \( n \), which in our case is 5.
• **The probability of success** \( p \), which represents the chance of someone being elderly. We've already calculated this as approximately 0.126.
• **The number of successes** \( k \), which is 4 in this problem.

To find the probability of exactly 4 successes (4 elderly individuals), we utilize the binomial probability formula, which combines these parameters into a straightforward calculation. This formula is incredibly useful whenever you have events following the yes/no binary pattern, exactly like coin flips or passing/failing a test.

Probability theory

Probability theory is the mathematical framework that allows us to quantify uncertainty. It gives us the tools to predict how likely events are to occur based on known data. Understanding probability means understanding how often something might logically happen. In this exercise, knowing the probability of any U.S. citizen being elderly was key.
Probability theory is built on some basic principles:

• Probabilities are numbers between 0 and 1, where 0 means the event will not happen, and 1 means it definitely will.
• The probability of all possible outcomes of an event adds up to 1.
• Probabilities help us measure the likelihood of complex events by breaking them into simpler components, using rules like addition and multiplication.

In the exercise, we started by calculating the probability that a single person is elderly (about 0.126) using these principles. From there, we applied more complex rules to find the probability of the situation involving 5 people.

Statistics

Statistics is the science of collecting, analyzing, and drawing conclusions from data. In everyday life, it helps translate numbers into insights. For this exercise, statistics helped us move beyond individual probabilities to interpret real-world scenarios involving multiple people.
Here's how statistics is applied here:

• **Data Collection**: We have a population size and a subgroup of elderly people.
• **Data Analysis**: We calculated the probability of interest (the chance that four of five people are elderly) using formulas from probability theory.
• **Drawing Conclusions**: We converted this probability to a percentage, providing a clear insight into likelihood. Converting numerical results into percentages makes them easier to interpret, especially in communication, because most people understand chances in percentage terms.

By using statistics, we took a situation about U.S. citizens and generalized it to understand broader patterns and likelihoods, showing how the numbers can tell a story about real-world phenomena.