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Probability theory is a branch of mathematics that studies the law of the number of random phenomena. Random phenomena are relative to decisive phenomena. A phenomenon in which a certain result necessarily occurs under certain conditions is called a decisive phenomenon. For example, at standard atmospheric pressure, water will boil when heated to 100 °C. Random phenomenon means that under the condition that the basic conditions are the same, before each test or observation, it is not certain which result will appear, showing contingency.
For example, if you throw a coin, you may have a front or back side. The realization of random phenomena and the observation of it are called random experiments. Each possible result of a randomized trial is called a basic event, and one or a set of basic events are collectively referred to as random events, or simply events. Typical randomized trials include craps, coin toss, playing cards, and roulette games.
Probability theory is a branch of mathematics related to probability. Although there are several different probabilistic interpretations, probability theory expresses it through a set of axioms, treating the concept in a rigorous mathematical way. Typically, these axioms normalize the probability probability space in one aspect, where a measure between the values 0 and 1 is assigned, called the probability measure, and the result of a pair is called the sample space. Any specified subset of these results is called an event.
The central body in probability theory consists of discrete and continuous random variables, probability distributions, and stochastic processes that provide mathematical abstractions of non-deterministic or uncertain processes or measures that can be single-shot or randomly evolved over time.