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Probability theory provides the fundamental foundation for predicting how likely an event is to occur. It's a mathematical framework that helps us reason with the uncertainties of random events. In our daily lives, probabilities are everywhere—from forecasting the weather to assessing risks. In statistics, probability theory helps us model randomness and variability in data, which is crucial for making predictions or inferences about populations based on sample data.

When dealing with events, probabilities are expressed as values between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. Every probabilistic experiment has outcomes that we can generally predict using probabilities. The sum of all possible outcomes' probabilities equals 1.

In relation to our problem, probability theory helps us calculate the likelihood of particular outcomes, such as how many people in a group sampled from the population have never traveled outside their home state.

Random sampling is a fundamental concept in probability and statistics. It involves selecting a subset of individuals from a larger population in such a way that every individual has an equal chance of being chosen. This method is often used to ensure that the sample is representative of the population, which is crucial for accurate statistical analysis.

By using random sampling, biases are minimized and the results can be generalized to the population as a whole. This concept is essential in surveys and experiments to maintain objectivity.

In the given exercise, a random sample of 80 Americans is chosen. This helps ensure that the conclusions drawn about the percentage of Americans who have never traveled outside their home state are representative of the wider population.

The complement rule is a useful tool in probability for simplifying calculations and understanding events. The rule states that the probability of an event occurring is equal to one minus the probability of its complement. The complement of an event is essentially anything that is not the event itself.

Mathematically, if the probability of an event happening is represented as \( P(A) \), then the probability of event A not happening (its complement) is: \[ P(A') = 1 - P(A) \]

This rule becomes particularly useful when calculating probabilities that involve the phrase "more than" or "less than" a certain number. In step 2 of the original solution, the complement rule helps calculate the probability of more than 12 Americans by finding the probability of 12 or fewer Americans, and then subtracting that from 1.

The binomial probability formula is essential for calculating the probability of a specific number of successes in a series of independent trials. This type of scenario is well-modeled by the binomial distribution, which applies when each trial has two possible outcomes.The formula is given by:\[ P(X=x) = \binom{n}{x} \times p^{x} \times q^{(n-x)} \]where

• \( n \) is the number of trials,
• \( x \) is the number of successes,
• \( p \) is the probability of success on a single trial,
• \( q = 1 - p \) is the probability of failure on a single trial.

In the context of the exercise, this formula is used to determine the probability that a certain number of Americans have never traveled outside their home state. By plugging in the values (such as \( n = 80 \) and \( p = 0.10 \)), one can calculate various probabilities by evaluating this formula for different values of \( x \). These calculations help find probabilities like more than 12, at least 12, and at most 12 Americans in the sample having never left their home state.