In probability theory, a probability space or a probability triple 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 .
A probability space consists of three elements: A sample space, , which is the set of all possible outcomes. An event space, which is a set of events, , an event being a set of outcomes in the sample space. A probability function, , 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 . 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 ("the die lands on 5"), as well as complex events such as ("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, would be mapped to , and would be mapped to .
When an experiment is conducted, we imagine that "nature" "selects" a single outcome, , from the sample space . All the events in the event space that contain the selected outcome 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 .
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.
Introduction
A probability space is a mathematical triplet that presents a model for a particular class of real-world situations. As with other models, its author ultimately defines which elements , , and will contain. The sample space is the set of all possible outcomes. An outcome is the result of a single execution of the model. Outcomes may be states of nature, possibilities, experimental results and the like. Every instance of the real-world situation (or run of the experiment) must produce exactly one outcome. If outcomes of different runs of an experiment differ in any way that matters, they are distinct outcomes. Which differences matter depends on the kind of analysis we want to do. This leads to different choices of sample space. The σ-algebra is a collection of all the events we would like to consider. This collection may or may not include each of the elementary events. Here, an "event" is a set of zero or more outcomes; that is, a subset of the sample space. An event is considered to have "happened" during an experiment when the outcome of the latter is an element of the event. Since the same outcome may be a member of many events, it is possible for many events to have happened given a single outcome. For example, when the trial consists of throwing two dice, the set of all outcomes with a sum of 7 pips may constitute an event, whereas outcomes with an odd number of pips may constitute another event. If the outcome is the element of the elementary event of two pips on the first die and five on the second, then both of the events, "7 pips" and "odd number of pips", are said to have happened. The probability measure is a set function returning an event's probability. A probability is a real number between zero (impossible events have probability zero, though probability-zero events are not necessarily impossible) and one (the event happens almost surely, with almost total certainty). Thus is a function The probability measure function must satisfy two simple requirements: First, the probability of a countable union of mutually exclusive events must be equal to the countable sum of the probabilities of each of these events. For example, the probability of the union of the mutually exclusive events and in the random experiment of one coin toss, , is the sum of probability for and the probability for , . Second, the probability of the sample space must be equal to 1 (which accounts for the fact that, given an execution of the model, some outcome must occur). In the previous example the probability of the set of outcomes must be equal to one, because it is entirely certain that the outcome will be either or (the model neglects any other possibility) in a single coin toss.