Probability space

Modelling a wheel of fortune using probability space

Probability space is a mathematical model used to describe scientific experiments. A probability space consists of three parts:

1. A sample space which lists all possible outcomes (where a sample space is often written as [math]\displaystyle{ \Omega }[/math], and an outcome as [math]\displaystyle{ \omega }[/math])
2. A set of events. Each event associates zero or more outcomes
3. A function that assigns probabilities to each event

An outcome is the result of a single execution of the model. Since individual outcomes might be of little practical use, more complex events are used to characterize groups of outcomes. The collection of all such events is a σ-algebra, sometimes written as [math]\displaystyle{ \mathcal{F} }[/math]. Finally, there is a need to specify each event's likelihood of happening. This is done using the probability measure function, P.

Once the probability space is established, one usually assumes that “nature” makes its move and selects a single outcome, ω, from the sample space Ω. All the events in [math]\displaystyle{ \mathcal{F} }[/math] that contain the selected outcome ω (recall that each event is a subset of Ω) are said to “have occurred”. The selection performed by nature is done in such a way that if the experiment were to be repeated an infinite number of times, then the relative frequencies of occurrence of each of the events would coincide with the probabilities described by the function P.

The prominent Soviet mathematician Andrey Kolmogorov introduced the notion of probability space, together with other axioms of probability, in the 1930s.

Probability Space Media

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  Probability space for throwing a die twice in succession: The sample space \Omega consists of all 36 possible outcomes; three different events (colored polygons) are shown, with their respective probabilities (assuming a discrete uniform distribution).

Related pages

• Random variable