Probability space

In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process (or “experiment”) consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind. One proposes that each time a situation of that kind arises, the set of possible outcomes is the same and the probabilities are also the same.

A probability space consists of three parts:

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 ${\displaystyle {\mathcal {F}}}$. Finally, there is a need to specify each event's likelihood of happening. This is done using the probability measure function, ${\displaystyle P}$.

Once the probability space is established, it is assumed that “nature” makes its move and selects a single outcome, ${\displaystyle \omega }$, from the sample space ${\displaystyle \Omega }$. All the events in ${\displaystyle {\mathcal {F}}}$ that contain the selected outcome ${\displaystyle \omega }$ (recall that each event is a subset of ${\displaystyle \Omega }$) 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, the relative frequencies of occurrence of each of the events would coincide with the probabilities prescribed by the function ${\displaystyle P}$.

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