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Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms .
Probability is the branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur. [note 1] [1] [2] This number is often expressed as a percentage (%), ranging from 0% to ...
The Dirac delta function, although not strictly a probability distribution, is a limiting form of many continuous probability functions. It represents a discrete probability distribution concentrated at 0 — a degenerate distribution — it is a Distribution (mathematics) in the generalized function sense; but the notation treats it as if it ...
The term law of total probability is sometimes taken to mean the law of alternatives, which is a special case of the law of total probability applying to discrete random variables. [ citation needed ] One author uses the terminology of the "Rule of Average Conditional Probabilities", [ 4 ] while another refers to it as the "continuous law of ...
More commonly, probability distributions are used to compare the relative occurrence of many different random values. Probability distributions can be defined in different ways and for discrete or for continuous variables. Distributions with special properties or for especially important applications are given specific names.
In probability theory, the chain rule [1] (also called the general product rule [2] [3]) describes how to calculate the probability of the intersection of, not necessarily independent, events or the joint distribution of random variables respectively, using conditional probabilities.
The probability is sometimes written to distinguish it from other functions and measure P to avoid having to define "P is a probability" and () is short for ({: ()}), where is the event space, is a random variable that is a function of (i.e., it depends upon ), and is some outcome of interest within the domain specified by (say, a particular ...
But if only a finite number of repetitions of the process are performed, different relative frequencies will appear in different series of trials. If these relative frequencies are to define the probability, the probability will be slightly different every time it is measured. But the real probability should be the same every time.