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The binomial distribution is a probability distribution that represents the number of successes in a fixed number of independent trials of a binary scenario. In simpler terms, imagine flipping a coin multiple times: each flip is an independent event with two possible outcomes (heads or tails). Similarly, when we survey American adult Internet users, each individual has two options: they either used the Internet for election information or they did not.

Regarding our exercise, the probability of success on a single trial is approximated to be 74% or 0.74, representing those who used the Internet for election-related information. The total number of trials is the number of users surveyed, which is 2,000. This scenario, when the number of trials is large, and the probability of success is not too close to 0 or 1, lends itself well to the normal approximation, which simplifies calculations significantly.

The Z-score is a statistical measure that tells us how many standard deviations an element is from the mean. In the context of our problem, if we want to find out how unusual or typical a certain value is, such as the number of users who used the Internet for election details, we compare this to what we'd expect — the mean (or average).

A Z-score can be positive or negative, indicating whether the value is above or below the mean, respectively. To calculate it, we subtract the mean from the value we're interested in and then divide by the standard deviation. It is a crucial part of finding probabilities as it allows us to compare different sets of data that may have different means and standard deviations.

A probability distribution is essentially a function that provides the probabilities of occurrence of different possible outcomes in an experiment. It quantifies the expected value of each outcome, which is critical in assessing risk and in making decisions under uncertainty.

In the normal approximation to the binomial distribution, we use a normal probability distribution as a stand-in for our binomial distribution because it is easier to work with, especially when dealing with large sample sizes. This normal distribution is characterized by its bell-shaped curve and is determined by two parameters: the mean and the standard deviation. It tells us how likely it is for different results to occur within our binomial experiment—like the number of users engaging with election content online.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means that most of the numbers are close to the mean (average). A high standard deviation indicates that the values are spread out over a wider range.

Standard deviation plays a key role in the calculation of Z-scores and, consequently, in the application of the normal approximation to binomial problems. To work out the standard deviation of our Internet users survey, for instance, we consider both the chance that someone would seek out election information (0.74) and the chance they wouldn't (0.26). The resulting standard deviation reflects how much the actual number of users could deviate from the expected average.