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Probability theory is the branch of mathematics that deals with the analysis of random events. The foundation of probability theory is the probability space, which consists of a set of possible outcomes, each with an associated probability that quantifies the chance of that outcome occurring.

When we perform an experiment like spinning a pointer, as mentioned in the original exercise, the possible outcomes are the numbers on which the pointer can land, and the corresponding probabilities are based on the probability space designed for that particular scenario.

Understanding the basic principles of probability such as events, outcomes, and associated probabilities, allows us to predict the likelihood of various outcomes and apply these concepts to solve practical problems – like computing the expected value.

The mathematical expectation, also known as the expected value, is the weighted average of all possible outcomes of a random variable, with the weights being their respective probabilities. This concept is incredibly useful in various fields such as economics, finance, and decision theory because it offers a single value that summarizes the distribution of outcomes.

The calculation of the expected value, as demonstrated in the step by step solution, involves multiplying each possible outcome by its probability and then summing up all these products. The result is a measure of the 'central tendency' or the 'long-run average' of the random variable after many repetitions of the experiment.

While the actual outcome of a random event may vary, the expected value provides a stable point of reference — like a forecast of what one can predict on average over time.

Discrete random variables are quantities that arise from random processes and have distinct and separate values. This is opposed to continuous random variables, which can take on any value within a range. In the context of the pointer example from the exercise, the numbers on which the pointer lands represent discrete outcomes because the pointer can only land on whole numbers like 1, 2, or 3, and nothing in between.

Each discrete outcome has a probability associated with it, which is a fundamental notion of discrete random variables. When we talk about the expected value of a discrete random variable, we're essentially asking: given all possible outcomes and their probabilities, what is the average outcome we would expect from an experiment if we were to repeat it an infinite number of times? It provides a way to predict outcomes of processes that are inherently uncertain and can be highly beneficial for making decisions based on statistical evidence.