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Understanding the concept of a probability distribution is fundamental to grasping the expected value calculation. A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. It's a way to map out the likelihood of each possible outcome and helps us to understand the behavior of random variables.

For discrete random variables, like in the exercise, a probability distribution is represented by a table, formula, or graph that links each outcome with its probability. The table provided in the exercise is a typical example of a probability distribution, showing clear values that the random variable, denoted as \(X\), can take along with the probabilities associated with these values. Probabilities are always non-negative and sum up to one. The completeness of this information ensures that we have a full view of how \(X\) behaves.

Outcome probabilities are the building blocks of any probability distribution. They quantify the chance that a specific event will occur. In our exercise, these probabilities are fractions that tell us how likely it is for the random variable \(X\) to equal a particular value. For instance, the probability of \(X\) being 10 is \(\frac{15}{50}\), or more simply, 30%.

It is essential to note that the sum of all outcome probabilities must equal 1 (or 100%) because one of the outcomes must occur. To improve your understanding of outcome probabilities, practice identifying and correctly interpreting these probabilities as they set the stage for further calculations, such as the expected value.

Mathematical expectation, commonly called the expected value, is the long-run average value of repetitions of the experiment it represents. It is a vital concept in statistics, representing the average outcome if an experiment is repeated many times. Simply put, it's what you would predict to happen on average over time.

To calculate the expected value, as shown in the exercise, you multiply each possible outcome by its probability and sum these products. The result is a single number that provides a sense of the central tendency of the probability distribution. In our exercise example, the expected value was 21, meaning if we repeat the process many times, the average value we would expect tends to be around 21.

This concept is used to make decisions in various fields, including economics and finance, where understanding the average expected payoff of different options helps to evaluate the risk and make well-informed choices.