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The binomial distribution is a common probability distribution used to model scenarios where there are two possible outcomes. In our exercise, when meeting a U.S. citizen, the outcomes are whether the person is elderly or not. This distribution is characterized by a set number of trials (such as meeting five citizens) and the chances of a particular outcome occurring (in this case, the probability of a citizen being elderly). The key to understanding binomial distribution is recognizing that each trial is independent. That means the age of one citizen does not affect the age of another. When situations like these occur, the binomial distribution helps us calculate the likelihood of a specific number of successful outcomes over the total trials. Here, it's about determining the probability that exactly four out of five citizens are elderly.

Once we establish the scenario fits a binomial model, we can proceed to probability calculations. The probability of an individual event occurring, such as meeting an elderly citizen, can be found by dividing the number of favorable cases by the total number of cases. For our exercise, this probability is calculated as \( P(\text{elderly}) = \frac{40}{317} \).Knowing the probability of not meeting an elderly person is equally important. This is done using the complementary probability: \( P(\text{not elderly}) = 1 - P(\text{elderly}) = 1 - \frac{40}{317} \).Armed with both probabilities, we can use them in our binomial probability formula. This involves multiplying the probability of four successful outcomes (elderly) with the probability of one unsuccessful outcome (not elderly), all while considering the number of combinations there are (choosing 4 elderly out of 5).

Understanding the demographics helps put the exercise into context. In 2013, about 40 million of the U.S.'s 317 million citizens were elderly, which is a significant portion of the populace. This data sets the stage for examining probabilities related to this demographic group.With this knowledge, we can quantify what it means in probabilistic terms to encounter elderly citizens. By using the ratio \( \frac{40}{317} \), students realize this calculation determines the likelihood of randomly picking an elderly individual among any citizen group. The exercise, through this probability, highlights the focus on this specific age group within statistical studies, making it more relatable and applicable.

Once the probabilities are calculated, they are often in a decimal format. To make the result more understandable, we convert these decimals into percentages. This transformation involves multiplying the decimal result by 100.For example, if the computed probability of meeting four elderly citizens is a decimal like 0.042, converting it into a percentage would mean \( 0.042 \times 100 = 4.2\% \). However, we are asked to round this to the nearest thousandth of a percent, which means expressing the percentage with three decimal places. This conversion is key in communicating probability findings clearly, such as telling someone there's a 4.2% chance of meeting four elderly citizens among five people.This concept of percentage conversion not only aids in comprehending the likelihoods but also helps in making the connection between statistical calculations and practical, real-world applications.