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Probability theory is a branch of mathematics concerned with analysing random phenomena and quantifying the likelihood of events occurring. The fundamental concept is the probability of an event, denoted as P(E), which is defined as a measure that falls within the range of 0 (impossibility) to 1 (certainty).

When solving probability problems, such as the dice-rolling exercise provided, it’s essential to understand that each result (or 'outcome') of a random experiment is an 'event', and the set of all possible outcomes is called the 'sample space'. In the example, rolling a 3 on one die is an event, while all possible outcomes when throwing the die constitute the sample space.

In practice, the probability of an event is calculated as the ratio of favorable outcomes – those outcomes that satisfy the event – to the total number of possible outcomes in the sample space. This is expressed in the formula: \[P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes in the sample space}}.\]

Independent events in probability refer to two or more events that do not influence the occurrence of each other. In other words, the outcome of one event has no effect on the outcome of the other. The classic schoolbook example of such events is the rolling of dice, as in the textbook exercise. Because the dice have no impact on each other, the events are considered independent.

Mathematically, two events E and F are independent if the probability of their intersection (both events occurring together) is equal to the product of their individual probabilities: \[ P(E \cap F) = P(E) \times P(F). \]

In the dice example, the realization that both E and F were independent comes by calculating their intersection's probability and confirming it equals the product of their separate probabilities. If the events affected each other, they’d be 'dependent', and we would see a different relationship between the probabilities of their intersection and the individual probabilities.

Finite Mathematics is an area of mathematics that focuses on various topics that have direct applications in fields such as business, engineering, social sciences, and life sciences. This can include, but is not limited to, subjects like algebra, matrices, mathematical models, statistics, and, notably, probability, which is the subject of our dice-rolling exercise. Finite mathematics deals with finite sets, countable events, and discrete quantities.

Infinite sets or continuous variables, typical of calculus, are not the focus of finite mathematics. Instead, it zeroes in on structured ways to count combinations and permutations, make decisions based on quantifiable information, and analyze situations like game theory, financial matters, or scheduling dilemmas. Understanding the underlying principles of probability as part of finite mathematics equips learners to approach a multitude of real-world problems with a mathematical perspective.