The probability theory and statistics is an important part of many of today’s human activities. Unfortunately, many have yet to appreciate the theory for what it is. The probability theory is a branch of mathematics that mainly deals with the possibilities in this world, the analysis of any random phenomena.

Random variables, events, and stochastic processes are all addressed within the realm of probability theory and statistics. These are mathematical abstractions based on non-deterministic events; they may also be measured quantities classified as either single occurrences, or they could also have evolved randomly. If you consider a roll of the dice as a random event, a resulting sequence of such random events when repeated will indeed exhibit some patterns that anyone can study and predict. This should tell you that two mathematical results that describe such patterns are considered the law regarding large numbers operating within the central limit theorem.

As the mathematical foundation for statistics, dealing with probability theories is vital in many human activities involving quantitative analysis consisting of significant data. The methods also apply to complex systems descriptions even if they only have partial knowledge of their current state, as is evident in statistical mechanics. The probabilistic nature that described the physical phenomena within atomic scales as outlined in the area of quantum mechanics is an important discovery in twentieth-century physics.

**Motivation**

Consider experiments that produce different outcomes. All the results, or rather the collection of results is referred to like the trial’s sample space. The existing power set of such sample space is formed upon by looking at the different collection of possible results. A good example would be the rolling of a die that may produce one among six possible results. A compilation of the possible results would automatically correspond to an odd number. This means that subsets (1, 3, or 5) are elements of the power sets within the sample spaces of the die rolls. We can now term such collections as “events,” with (1, 3, and 5) as the event that the die roll will fall on an odd number.

**How Probability Theory and Statistics Came to Be**

The probability theory credits its roots in the attempts made to analyze the games of a chance back in the Sixteenth and Seventeenth century. During the days, it primarily considered discrete events, where the methods used were combinatorial. Analytical considerations eventually directed continuous variable incorporation into the theory.

Such direction culminated in what we all know today as the modern probability theory, which combined the sample space notion and measure theory. Both then paved the way for the axiom system dedicated to the mechanics of the theory that first surfaced back in the thirties. The development quickly became an undisputed axiomatic basis for the modern probability theory even though there are alternatives that anyone can take up, particularly the adoption of finite aspects rather than the countable additivity.

It must be noted that most introductions to the probability theory and statistics address continuous and discrete probability distributions separately. Mathematically advanced measure theories based on the treatment of probabilities cover the continuous or discrete, or mix.