Probability space

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 of a random process under consideration.
An event space,, which is a set of events, where an event is a subset of outcomes in the sample space.
A probability function,, which assigns, to each event in the event space, a probability, which is a number between 0 and 1.

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 is typically the set where each element in the set is a label which represents the outcome of the die landing on that label. For example, represents the outcome that the die lands on 1.
The event space could be the set of all subsets of the sample space, which would then contain simple events such as , as well as complex events such as .
The probability function would then 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, it results in exactly one outcome from the sample space. All the events in the event space that contain the selected outcome are said to "have occurred". The probability function 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.

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 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 and one. 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. 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 in a single coin toss.

Not every subset of the sample space must necessarily be considered an event: some of the subsets are simply not of interest, others cannot be "measured". This is not so obvious in a case like a coin toss. In a different example, one could consider javelin throw lengths, where the events typically are intervals like "between 60 and 65 meters" and unions of such intervals, but not sets like the "irrational numbers between 60 and 65 meters".

Definition

In short, a probability space is a measure space such that the measure of the whole space is equal to one.
The expanded definition is the following: a probability space is a triple consisting of:

the sample space – an arbitrary non-empty set,
the σ-algebra – a set of subsets of, called events, such that:
* contains the sample space:,
* is closed under complements: if, then also,
* is closed under countable unions: if for, then also
** The corollary from the previous two properties and De Morgan's law is that is also closed under countable intersections: if for, then also
the probability measure – a function on such that:
* P is countably additive : if is a countable collection of pairwise disjoint sets, then
* the measure of the entire sample space is equal to one:.
Discrete case

needs only at most countable sample spaces. Probabilities can be ascribed to points of by the probability mass function such that. All subsets of can be treated as events. The probability measure takes the simple form
The greatest σ-algebra describes the complete information. In general, a σ-algebra corresponds to a finite or countable partition, the general form of an event being. See also the examples.
The case is permitted by the definition, but rarely used, since such can safely be excluded from the sample space.

General case

If is uncountable, still, it may happen that for some ; such are called atoms. They are an at most countable set, whose probability is the sum of probabilities of all atoms. If this sum is equal to 1 then all other points can safely be excluded from the sample space, returning us to the discrete case. Otherwise, if the sum of probabilities of all atoms is between 0 and 1, then the probability space decomposes into a discrete part and a non-atomic part.

Non-atomic case

If for all , then equation fails: the probability of a set is not necessarily the sum over the probabilities of its elements, as summation is only defined for countable numbers of elements. This makes the probability space theory much more technical. A formulation stronger than summation, measure theory is applicable. Initially the probabilities are ascribed to some "generator" sets. Then a limiting procedure allows assigning probabilities to sets that are limits of sequences of generator sets, or limits of limits, and so on. All these sets are the σ-algebra. For technical details see Carathéodory's extension theorem. Sets belonging to are called measurable. In general they are much more complicated than generator sets, but much better than non-measurable sets.

Complete probability space

A probability space is said to be a complete probability space if for all with and all one has. Often, the study of probability spaces is restricted to complete probability spaces.

Examples

Discrete examples

Example 1

If the experiment consists of just one flip of a fair coin, then the outcome is either heads or tails:. The σ-algebra contains events, namely: , , , and ; in other words,. There is a fifty percent chance of tossing heads and fifty percent for tails, so the probability measure in this example is,,,.

Example 2

The fair coin is tossed three times. There are 8 possible outcomes: . The complete information is described by the σ-algebra of events, where each of the events is a subset of Ω.
Alice knows the outcome of the second toss only. Thus her incomplete information is described by the partition, where ⊔ is the disjoint union, and the corresponding σ-algebra. Bryan knows only the total number of tails. His partition contains four parts: ; accordingly, his σ-algebra contains 2⁴ = 16 events.
The two σ-algebras are incomparable: neither nor ; both are sub-σ-algebras of 2^Ω.

Example 3

If 100 voters are to be drawn randomly from among all voters in California and asked whom they will vote for governor, then the set of all sequences of 100 Californian voters would be the sample space Ω. We assume that sampling without replacement is used: only sequences of 100 different voters are allowed. For simplicity an ordered sample is considered, that is a sequence is different from. We also take for granted that each potential voter knows exactly his/her future choice, that is he/she does not choose randomly.
Alice knows only whether or not Arnold Schwarzenegger has received at least 60 votes. Her incomplete information is described by the σ-algebra that contains: the set of all sequences in Ω where at least 60 people vote for Schwarzenegger; the set of all sequences where fewer than 60 vote for Schwarzenegger; the whole sample space Ω; and the empty set ∅.
Bryan knows the exact number of voters who are going to vote for Schwarzenegger. His incomplete information is described by the corresponding partition and the σ-algebra consists of 2¹⁰¹ events.
In this case, Alice's σ-algebra is a subset of Bryan's:. Bryan's σ-algebra is in turn a subset of the much larger "complete information" σ-algebra 2^Ω consisting of events, where n is the number of all potential voters in California.