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Calculating probability in a binomial distribution scenario involves understanding how likely it is for a given number of outcomes ("successes") to occur in a fixed number of trials. In our exercise, we are interested in various probabilities related to households having only cell phone service. To compute these probabilities, we utilize a mathematical formula from the binomial distribution model:

• First, determine the probability of success in a single trial, noted as \( p \). In this case, \( p = 0.52 \) or 52%.
• The number of trials is \( n = 40 \) households.
• For each specific question, we focus on a particular number of successes \( x \), such as exactly 20 or fewer than 20, etc.
• Use the probability formula: \( P(x) = C(n, x) \cdot p^x \cdot (1-p)^{n-x} \).

Substitute \( n \), \( p \), and \( x \) into the formula to find the specific probability. This formula helps involve each potential combination of successes among the trials, by using the probability of success powered by the number of desired successes and multiplied by the probability of failure accordingly.

A statistical distribution is a method to describe all the potential outcomes of a random variable. In this exercise, we focus on the binomial distribution, which is a common type of statistical distribution ideal for scenarios involving multiple trials with two possible outcomes - success or failure.The binomial distribution is characterized by two main parameters: the number of trials \( n \) and the probability of success in a single trial \( p \). It models the number of successes \( x \) in \( n \) trials.

• In our problem, the trials are the number of different households sampled. Each trial represents one household.
• The outcomes concern whether each household only has cell phone service (success) or not (failure).
• The binomial distribution accurately helps us predict the likelihood of observing a certain number of successes within our sample.

This kind of distribution is widely used when dealing with probabilities across different scenarios ranging from scientific studies to business modeling, wherever decisions rely on understanding frequencies and magnitudes of discrete events.

The combination formula is a key component in calculating probabilities for binomial distributions. It helps determine how many different ways a specific outcome can occur.The formula for combinations, represented as \( C(n, x) \), calculates the total number of ways \( x \) successes can be achieved in \( n \) trials. The formula is given by:\[C(n, x) = \frac{n!}{x!(n-x)!}\]

• Here, \( n! \) ("n factorial") means multiplying all whole numbers from \( n \) down to 1.
• For the specific number of successes \( x \), \( x! \) works similarly, and \((n-x)!\) accounts for the rest of the trials.
• This formula is crucial as it ensures all combinations of outcomes are considered, making the probability calculation complete.

In practical terms, the combination formula lets you see how many different ways you can pick \( x \) successful trials from \( n \), which is foundational for applying these results in the broader probability equation for your binomial distribution exercise.