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An event space, , which is a set of events, where an event is a subset of outcomes in the sample space. A probability function , P {\displaystyle P} , which assigns, to each event in the event space, a probability , which is a number between 0 and 1 (inclusive).
An event, however, is any subset of the sample space, including any singleton set (an elementary event), the empty set (an impossible event, with probability zero) and the sample space itself (a certain event, with probability one). Other events are proper subsets of the sample space that contain multiple elements. So, for example, potential ...
So, the probability of the entire sample space is 1, and the probability of the null event is 0. The function f ( x ) {\displaystyle f(x)\,} mapping a point in the sample space to the "probability" value is called a probability mass function abbreviated as pmf .
The assumptions as to setting up the axioms can be summarised as follows: Let (,,) be a measure space with () being the probability of some event, and () =. Then ( Ω , F , P ) {\displaystyle (\Omega ,F,P)} is a probability space , with sample space Ω {\displaystyle \Omega } , event space F {\displaystyle F} and probability measure P ...
In an elementary approach to probability, any subset of the sample space is usually called an event. [9] However, this gives rise to problems when the sample space is continuous, so that a more precise definition of an event is necessary.
A probability measure mapping the σ-algebra for events to the unit interval.. The requirements for a set function to be a probability measure on a σ-algebra are that: . must return results in the unit interval [,], returning for the empty set and for the entire space.
The event that contains all possible outcomes of an experiment is its sample space. A single outcome can be a part of many different events. [4] Typically, when the sample space is finite, any subset of the sample space is an event (that is, all elements of the power set of the sample space are defined as
A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often represented in notation by Ω , {\displaystyle \ \Omega \ ,} is the set of all possible outcomes of a random phenomenon being observed.