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In understanding probability distribution, it is essential to grasp the concept of a random variable. A random variable is a numerical description of the outcome of a statistical experiment. In simpler terms, it is a variable whose possible values are numerical outcomes of a random phenomenon.

There are two types of random variables: discrete and continuous. A discrete random variable has a countable number of possible values, like the result of rolling a die (1, 2, 3, 4, 5, 6). On the other hand, a continuous random variable has an infinite number of possible values within a given range, such as the exact height of students in a school.

Each random variable has an associated probability distribution that provides the probabilities of its potential values. In our exercise, the random variable 'x' can take on the values 0, 1, 2, 3, or 4, which are discrete outcomes. The function p(x) maps each outcome with its probability, fundamentally outlining the fabric of probability theory.

Probability theory is the branch of mathematics that deals with the analysis of random events. The core of the theory lies in quantifying the likelihood of various outcomes. This ranges from the probability of a simple event, like flipping a coin, to the more complex, such as forecasting weather patterns.

An understanding of probability theory aids in comprehending events that do not have a deterministic outcome. It's fundamentally based on set theory and involves elements like sample spaces, events, and the axioms of probability. The theory asserts that the sum of probabilities of all possible outcomes in a sample space is equal to 1.

In the context of the textbook exercise, probability theory guides us through the process of determining the probability of the random variable 'x' taking on the value 3. By understanding that the probabilities of all possible outcomes must add up to 1, we applied this knowledge to find that missing value.

Statistical probability refers to the use of probability theory to analyze statistical experiments, where the outcome is not deterministic but is subject to chance. This facet of mathematics is crucial in predicting the behavior of systems and making informed decisions based on likelihoods.

Statistical probability encompasses collecting data, summarizing it, and making inferences about a population based on sampled data. When we looked at the probability distribution table in our exercise, we were essentially engaging with statistical probability. We had a set of outcomes (x) with their respective probabilities which, when correctly determined, should adhere to the rules of statistical probability, such as all probabilities adding up to 1.

By following this rule, we resolved the textbook problem. However, an important practical takeaway is that understanding statistical probability can significantly enhance the analysis and interpretation of real-world phenomena, from genetics to finance and beyond.