Probability space

Part of a series on statistics
Probability theory
Probability Axioms Determinism System Indeterminism Randomness
Probability space Sample space Event Collectively exhaustive events Elementary event Mutual exclusivity Outcome Singleton Experiment Bernoulli trial Probability distribution Bernoulli distribution Binomial distribution Exponential distribution Normal distribution Pareto distribution Poisson distribution Probability measure Random variable Bernoulli process Continuous or discrete Expected value Variance Markov chain Observed value Random walk Stochastic process
Complementary event Joint probability Marginal probability Conditional probability
Independence Conditional independence Law of total probability Law of large numbers Bayes' theorem Boole's inequality
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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, 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,

1. The sample space ${\displaystyle \Omega }$ is typically the set ${\displaystyle \{1,2,3,4,5,6\}}$ where each element in the set is a label which represents the outcome of the die landing on that label. For example, ${\displaystyle 1}$ represents the outcome that the die lands on 1.
2. The event space ${\displaystyle {\mathcal {F}}}$ could be 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").
3. The probability function ${\displaystyle P}$ would then 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, it results in exactly one 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". The probability function ${\displaystyle P}$ 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.