# Probability space

## Mathematical concept / From Wikipedia, the free encyclopedia

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In probability theory, a **probability space** or a **probability triple** $(\Omega ,{\mathcal {F}},P)$ 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]}

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

In order to provide a model of probability, these elements must satisfy probability axioms.

In the example of the throw of a standard die,

- The sample space $\Omega$ is typically the set $\{1,2,3,4,5,6\}$ where each element in the set is a label which represents the outcome of the die landing on that label. For example, $1$ represents the outcome that the die lands on 1.
- The event space ${\mathcal {F}}$ could be the set of all subsets of the sample space, which would then contain simple events such as $\{5\}$ ("the die lands on 5"), as well as complex events such as $\{2,4,6\}$ ("the die lands on an even number").
- The probability function $P$ would then map each event to the number of outcomes in that event divided by 6 – so for example, $\{5\}$ would be mapped to $1/6$, and $\{2,4,6\}$ would be mapped to $3/6=1/2$.

When an experiment is conducted, it results in exactly one outcome $\omega$ from the sample space $\Omega$. All the events in the event space ${\mathcal {F}}$ that contain the selected outcome $\omega$ are said to "have occurred". The probability function $P$ must be so defined that if the experiment were repeated arbitrarily many times, the number of occurrences of each event as a fraction of the total number of experiments, will most likely tend towards the probability assigned to that event.

The Soviet mathematician Andrey Kolmogorov introduced the notion of a probability space and the axioms of probability in the 1930s. In modern probability theory, there are alternative approaches for axiomatization, such as the algebra of random variables.