# Probability space

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In probability theory, a probability space or a probability triple ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a die.

A probability space consists of three elements:[1][2]

1. A sample space, ${\displaystyle \Omega }$, which is the set of all possible outcomes.
2. An event space, which is a set of events, ${\displaystyle {\mathcal {F}}}$, an event being a set of outcomes in the sample space.
3. A probability function, ${\displaystyle P}$, which assigns each event in the event space a probability, which is a number between 0 and 1.

In order to provide a sensible model of probability, these elements must satisfy a number of axioms, detailed in this article.

In the example of the throw of a standard die, we would take the sample space to be ${\displaystyle \{1,2,3,4,5,6\}}$. For the event space, we could simply use the set of all subsets of the sample space, which would then contain simple events such as ${\displaystyle \{5\}}$ ("the die lands on 5"), as well as complex events such as ${\displaystyle \{2,4,6\}}$ ("the die lands on an even number"). Finally, for the probability function, we would map each event to the number of outcomes in that event divided by 6 – so for example, ${\displaystyle \{5\}}$ would be mapped to ${\displaystyle 1/6}$, and ${\displaystyle \{2,4,6\}}$ would be mapped to ${\displaystyle 3/6=1/2}$.

When an experiment is conducted, we imagine that "nature" "selects" a single outcome, ${\displaystyle \omega }$, from the sample space ${\displaystyle \Omega }$. All the events in the event space ${\displaystyle {\mathcal {F}}}$ that contain the selected outcome ${\displaystyle \omega }$ are said to "have occurred". This "selection" happens in such a way that if the experiment were repeated many times, the number of occurrences of each event, as a fraction of the total number of experiments, would most likely tend towards the probability assigned to that event by the probability function ${\displaystyle P}$.

The Soviet mathematician Andrey Kolmogorov introduced the notion of probability space, together with other axioms of probability, in the 1930s. In modern probability theory there are a number of alternative approaches for axiomatization – for example, algebra of random variables.