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| {{About|mathematical term|the novel|Probability Space (novel)}}
| | I'm Uwe and I live in a seaside city in northern Italy, Chesallet Sarre. I'm 26 and I'm will soon finish my study at Comparative Politics.<br><br>Also visit my weblog kitchen renovation ideas ([http://www.homeimprovementdaily.com Recommended Browsing]) |
| {{More footnotes|date=September 2009}}
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| {{Probability fundamentals}}
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| In [[probability theory]], a '''probability space''' or a '''probability triple''' is a [[space (mathematics)|mathematical construct]] that models a real-world process (or "experiment") consisting of states that occur [[randomness|randomly]]. A probability space is constructed with a specific kind of situation or experiment in mind. One proposes that each time a situation of that kind arises, the set of possible outcomes is the same and the probabilities are also the same.
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| A probability space consists of three parts:
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| # A [[sample space]], Ω, which is the set of all possible outcomes.
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| # A set of [[event (probability theory)|event]]s <math>\scriptstyle \mathcal{F}</math>, where each event is a set containing zero or more outcomes.
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| # The assignment of [[probability|probabilities]] to the events; that is, a function ''P'' from events to probabilities.
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| An outcome is the result of a single execution of the model. Since individual outcomes might be of little practical use, more complex ''events'' are used to characterize groups of outcomes. The collection of all such events is a ''[[σ-algebra]]'' <math>\scriptstyle \mathcal{F}</math>. Finally, there is a need to specify each event's likelihood of happening. This is done using the ''[[probability measure]]'' function, ''P''.
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| Once the probability space is established, it is assumed that “nature” makes its move and selects a single outcome, ''ω'', from the sample space Ω. All the events in <math>\scriptstyle \mathcal{F}</math> that contain the selected outcome ''ω'' (recall that each event is a subset of Ω) are said to “have occurred”. The selection performed by nature is done in such a way that if the experiment were to be repeated an infinite number of times, the relative frequencies of occurrence of each of the events would coincide with the probabilities prescribed by the function ''P''.
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| The prominent Soviet mathematician [[Andrey Kolmogorov]] introduced the notion of probability space, together with other [[axioms of probability]], in the 1930s. Nowadays alternative approaches for axiomatization of probability theory exist; see “[[Algebra of random variables]]”, for example.
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| This article is concerned with the mathematics of manipulating probabilities. The article [[probability interpretations]] outlines several alternative views of what "probability" means and how it should be interpreted. In addition, there have been attempts to construct theories for quantities that are notionally similar to probabilities but do not obey all their rules; see, for example, [[Free probability]], [[Fuzzy logic]], [[Possibility theory]], [[Negative probability]] and [[Quantum probability]].
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| ==Introduction==
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| {{cleanup|section|date=September 2009}}
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| A probability space is a mathematical triplet (Ω, <math>\scriptstyle \mathcal{F}</math>, ''P'') that
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| presents a [[mathematical model|model]] for a particular class of real-world situations.
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| As with other models, its author ultimately defines which elements Ω, <math>\scriptstyle \mathcal{F}</math>, and ''P'' will contain.
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| * The [[sample space]] Ω is a set of outcomes. An [[Outcome (probability)|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. What differences matter depends, of course, on the kind of analysis we want to do. This leads to different choices of sample space.
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| * The [[σ-algebra]] <math>\scriptstyle \mathcal{F}</math> is a collection of all and only [[event (probability theory)|event]]s (not necessarily [[Elementary event|elementary]]) we would like to consider. Here, an "event" is a set of zero or more outcomes, i.e., a [[subset]] of the sample space. An event is considered to have "happened" when the outcome is a member 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 of "7 pips" and "odd number of pips" have also happened.
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| * The [[probability measure]] ''P'' is a function returning an event's [[probability]]. A probability is a real number between zero (impossible events have probability zero, though probability-zero events need not be impossible) and one (the event happens [[almost surely]]). Thus ''P'' is a function <math>\scriptstyle P:\ \mathcal{F} \rightarrow [0,1]</math>. The probability measure function must satisfy a simple requirement: the probability of a union of two (or [[countable set|countably many]]) disjoint events must be equal to the sum of probabilities of each of these events. For example, if two events are ''Heads'' and ''Tails'', then the probability of ''Heads-or-Tails'' must be equal to the sum of probabilities for ''Heads'' and ''Tails'').
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| Not every subset of the sample space Ω 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 "irrational numbers between 60 and 65 meters"
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| ==Definition==
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| In short, a probability space is a [[measure (mathematics)|measure space]] such that the measure of the whole space is equal to one.
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| The expanded definition is following: a probability space is a triple <math style="position:relative;top:-.1em">\scriptstyle (\Omega,\; \mathcal{F},\; P)</math> consisting of:
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| * the [[sample space]] Ω — an arbitrary [[non-empty set]],
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| * the [[sigma-algebra|σ-algebra]] <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> ⊆ 2<sup>Ω</sup> (also called σ-field) — a set of subsets of Ω, called [[event (probability theory)|event]]s, such that:
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| ** <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> contains the empty set: <math style="position:relative;top:-.2em">\scriptstyle \emptyset\in \mathcal{F}</math>,
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| ** <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> is closed under [[complement (set theory)|complement]]s: if ''A''∈<math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math>, then also (Ω∖''A'')∈<math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math>,
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| ** <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> is closed under [[countable set|countable]] [[Union (set theory)|unions]]: if ''A<sub>i</sub>''∈<math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> for ''i''=1,2,..., then also (∪<sub>''i''</sub>''A<sub>i</sub>'')∈<math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math>
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| *** The corollary from the previous two properties and [[De Morgan’s law]] is that <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> is also closed under countable [[Intersection (set theory)|intersections]]: if ''A<sub>i</sub>''∈<math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> for ''i''=1,2,..., then also (∩<sub>''i''</sub>''A<sub>i</sub>'')∈<math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math>
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| * the [[probability measure]] ''P'': <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math>→[0,1] — a function on <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> such that:
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| ** ''P'' is [[countably additive]]: if {''A<sub>i</sub>''}⊆<math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> is a countable collection of pairwise [[disjoint sets]], then ''P''(⊔''A<sub>i</sub>'') = ∑''P''(''A<sub>i</sub>''), where “⊔” denotes the [[disjoint union]],
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| ** the measure of entire sample space is equal to one: ''P''(Ω) = 1.
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| ==Discrete case==
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| Discrete probability theory needs only [[countable set|at most countable]] sample spaces Ω. Probabilities can be ascribed to points of Ω by the [[probability mass function]] ''p'': Ω→[0,1] such that ∑<sub>''ω''∈Ω</sub> ''p''(''ω'') = 1. All subsets of Ω can be treated as events (thus, <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> = 2<sup>Ω</sup> is the [[power set]]). The probability measure takes the simple form
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| <math>
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| (*) \qquad P(A) = \sum_{\omega\in A} p(\omega) \quad \text{for all } A \subseteq \Omega \, .
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| </math>
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| The greatest σ-algebra <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> = 2<sup>Ω</sup> describes the complete information. In general, a σ-algebra <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> ⊆ 2<sup>Ω</sup> corresponds to a finite or countable [[partition of a set|partition]] Ω = ''B''<sub>1</sub> ⊔ ''B''<sub>2</sub> ⊔ ..., the general form of an event ''A'' ∈ <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> being ''A'' = ''B''<sub>''k''<sub>1</sub></sub> ⊔ ''B''<sub>''k''<sub>2</sub></sub> ⊔ ... (here ⊔ means the [[disjoint union]].) See also the examples.
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| The case ''p''(''ω'') = 0 is permitted by the definition, but rarely used, since such ''ω'' can safely be excluded from the sample space.
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| ==General case==
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| If Ω is [[uncountable set|uncountable]], still, it may happen that ''p''(''ω'') ≠ 0 for some ''ω''; such ''ω'' are called [[atom (measure theory)|atom]]s. They are an at most countable (maybe, [[empty set|empty]]) 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 less than 1 (maybe 0), then the probability space decomposes into a discrete (atomic) part (maybe empty) and a [[atom (measure theory)|non-atomic]] part.
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| ==Non-atomic case==
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| If ''p''(''ω'') = 0 for all ''ω''∈Ω (in this case, Ω must be uncountable, because otherwise P(Ω)=1 could not be satisfied), then equation (∗) fails: the probability of a set is not the sum over its elements, as summation is only defined for countable amount 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 (see the examples). 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 <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math>. For technical details see [[Carathéodory's extension theorem]]. Sets belonging to <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> are called [[measurable]]. In general they are much more complicated than generator sets, but much better than [[non-measurable set]]s.
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| ==Complete probability space==
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| A probability space <math style="position:relative;top:-.1em">\scriptstyle (\Omega,\; \mathcal{F},\; P)</math> is said to be a complete probability space if for all <math style="position:relative;top:-.1em">\scriptstyle B\, \in \,\mathcal{F}</math> with <math style="position:relative;top:-.1em">\scriptstyle P(B)\,=\;0 </math> and all <math style="position:relative;top:-.1em">\scriptstyle A\; \subset \;B </math> one has <math style="position:relative;top:-.1em">\scriptstyle A \;\in\; \mathcal{F}</math>. Often, the study of probability spaces is restricted to complete probability spaces.
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| ==Examples==
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| ===Discrete examples===
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| ====Example 1====
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| If the experiment consists of just one flip of a perfect coin, then the outcomes are either heads or tails: Ω = {H, T}. The σ-algebra <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> = 2<sup>Ω</sup> contains 2² = 4 events, namely: {H} – “heads”, {T} – “tails”, {} – “neither heads nor tails”, and {H,T} – “either heads or tails”. So, <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> = <nowiki>{{</nowiki>}, {H}, {T}, {H,T}}. There is a fifty percent chance of tossing heads, and fifty percent for tails. Thus the probability measure in this example is ''P''({}) = 0, ''P''({H}) = 0.5, ''P''({T}) = 0.5, ''P''({H,T}) = 1.
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| ====Example 2====
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| The fair coin is tossed three times. There are 8 possible outcomes: Ω = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} (here “HTH” for example means that first time the coin landed heads, the second time tails, and the last time heads again). The complete information is described by the σ-algebra <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> = 2<sup>Ω</sup> of 2<sup>8</sup> = 256 events, where each of the events is a subset of Ω.
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| Alice knows the outcome of the second toss only. Thus her incomplete information is described by the partition Ω = A<sub>1</sub> ⊔ A<sub>2</sub> = {HHH, HHT, THH, THT} ⊔ {HTH, HTT, TTH, TTT}, and the corresponding σ-algebra <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math><sub>Alice</sub> = <nowiki>{{</nowiki>}, A<sub>1</sub>, A<sub>2</sub>, Ω}. Brian knows only the total number of tails. His partition contains four parts: Ω = B<sub>0</sub> ⊔ B<sub>1</sub> ⊔ B<sub>2</sub> ⊔ B<sub>3</sub> = {HHH} ⊔ {HHT, HTH, THH} ⊔ {TTH, THT, HTT} ⊔ {TTT}; accordingly, his σ-algebra <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math><sub>Brian</sub> contains 2<sup>4</sup> = 16 events.
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| The two σ-algebras are [[comparability|incomparable]]: neither <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math><sub>Alice</sub> ⊆ <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math><sub>Brian</sub> nor <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math><sub>Brian</sub> ⊆ <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math><sub>Alice</sub>; both are sub-σ-algebras of 2<sup>Ω</sup>.
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| ====Example 3====
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| 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 [[sequence]]s of 100 Californian voters would be the sample space Ω. We assume that [[simple random sample|sampling without replacement]] is used: only sequences of 100 ''different'' voters are allowed. For simplicity an ordered sample is considered, that is a sequence {Alice, Brian} is different from {Brian, Alice}. We also take for granted that each potential voter knows exactly his future choice, that is he/she doesn’t choose randomly.
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| Alice knows only whether or not [[Arnold Schwarzenegger]] has received at least 60 votes. Her incomplete information is described by the σ-algebra <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math><sub>Alice</sub> that contains: (1) the set of all sequences in Ω where at least 60 people vote for Schwarzenegger; (2) the set of all sequences where fewer than 60 vote for Schwarzenegger; (3) the whole sample space Ω; and (4) the empty set ∅.
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| Brian knows the exact number of voters who are going to vote for Schwarzenegger. His incomplete information is described by the corresponding partition Ω = B<sub>0</sub> ⊔ B<sub>1</sub> ... ⊔ B<sub>100</sub> (though some of these sets may be empty, depending on the Californian voters...) and the σ-algebra <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math><sub>Brian</sub> consists of 2<sup>101</sup> events.
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| In this case Alice’s σ-algebra is a subset of Brian’s: <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math><sub>Alice</sub> ⊂ <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math><sub>Brian</sub>. The Brian’s σ-algebra is in turn the subset of the much larger “complete information” σ-algebra 2<sup>Ω</sup> consisting of {{nobreak|2<sup>''n''(''n''−1)...(''n''−99)</sup>}} events, where ''n'' is the number of all potential voters in California.
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| ===Non-atomic examples===
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| ====Example 4====
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| A number between 0 and 1 is chosen at random, uniformly. Here Ω = [0,1], <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> is the σ-algebra of [[Borel set]]s on Ω, and ''P'' is the [[Lebesgue measure]] on [0,1].
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| In this case the open intervals of the form (''a'',''b''), where 0<''a''<''b''<1, could be taken as the generator sets. Each such set can be ascribed the probability of ''P''((''a'',''b'')) = (''b''−''a''), which generates the [[Lebesgue measure]] on [0,1], and the [[Borel σ-algebra]] on Ω.
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| ====Example 5====
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| A fair coin is tossed endlessly. Here one can take Ω = {0,1}<sup>∞</sup>, the set of all infinite sequences of numbers 0 and 1. [[Cylinder set]]s {(''x''<sub>1</sub>,''x''<sub>2</sub>,...)∈Ω: ''x''<sub>1</sub>=''a''<sub>1</sub>, ..., ''x''<sub>''n''</sub>=''a''<sub>''n''</sub>} may be used as the generator sets. Each such set describes an event in which the first ''n'' tosses have resulted in a fixed sequence (''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>), and the rest of the sequence may be arbitrary. Each such event can be naturally given the probability of 2<sup>−''n''</sup>.
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| These two non-atomic examples are closely related: a sequence (''x''<sub>1</sub>,''x''<sub>2</sub>,...) ∈ {0,1}<sup>∞</sup> leads to the number 2<sup>−1</sup>''x''<sub>1</sub> + 2<sup>−2</sup>''x''<sub>2</sub> + ... ∈ [0,1]. This is not a [[one-to-one correspondence]] between {0,1}<sup>∞</sup> and [0,1] however: it is an [[standard probability space|isomorphism modulo zero]], which allows for treating the two probability spaces as two forms of the same probability space. In fact, all non-pathologic non-atomic probability spaces are the same in this sense. They are so-called [[standard probability space]]s. Basic applications of probability spaces are insensitive to standardness. However, non-discrete conditioning is easy and natural on standard probability spaces, otherwise it becomes obscure.
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| ==Related concepts==
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| ===Probability distribution===
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| Any [[probability distribution]] defines a probability measure.
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| ===Random variables===
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| A [[random variable]] ''X'' is a [[measurable function]] ''X'': Ω→''S'' from the sample space Ω to another measurable space ''S'' called the ''state space''.
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| The notation Pr(''X''∈''A'') is a commonly used shorthand for ''P''({''ω''∈Ω: ''X''(''ω'')∈''A''}).
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| ===Defining the events in terms of the sample space===
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| If Ω is [[countable]] we almost always define <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> as the [[power set]] of Ω, i.e. <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> = 2<sup>Ω</sup> which is trivially a σ-algebra and the biggest one we can create using Ω. We can therefore omit ℱ and just write (Ω,P) to define the probability space.
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| On the other hand, if Ω is [[uncountable]] and we use <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> = 2<sup>Ω</sup> we get into trouble defining our probability measure ''P'' because <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math> is too “large”, i.e. there will often be sets to which it will be impossible to assign a unique measure, giving rise to problems like the [[Banach–Tarski paradox]]. In this case, we have to use a smaller σ-algebra <math style="position:relative;top:-.2em">\scriptstyle \mathcal{F}</math>, for example the [[Borel algebra]] of Ω, which is the smallest σ-algebra that makes all open sets measurable.
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| ===Conditional probability===
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| Kolmogorov’s definition of probability spaces gives rise to the natural concept of [[conditional probability]]. Every set ''A'' with non-zero probability (that is, ''P''(''A'') > 0) defines another probability measure
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| : <math>
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| P(B | A) = {P(B \cap A) \over P(A)}
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| </math>
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| on the space. This is usually pronounced as the “probability of ''B'' given ''A''”.
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| For any event ''B'' such that ''P''(''B'') > 0 the function ''Q'' defined by ''Q''(''A'') = ''P''(''A''|''B'') for all events ''A'' is itself a probability measure.
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| ===Independence===
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| Two events, ''A'' and ''B'' are said to be [[Statistical independence|independent]] if ''P''(''A''∩''B'')=''P''(''A'')''P''(''B'').
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| Two random variables, ''X'' and ''Y'', are said to be independent if any event defined in terms of ''X'' is independent of any event defined in terms of ''Y''. Formally, they generate independent σ-algebras, where two σ-algebras ''G'' and ''H'', which are subsets of ''F'' are said to be independent if any element of ''G'' is independent of any element of ''H''.
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| ===Mutual exclusivity===
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| Two events, ''A'' and ''B'' are said to be [[mutually exclusive]] or ''disjoint'' if ''P''(''A''∩''B'') = 0. (This is weaker than ''A''∩''B'' = ∅, which is the definition of [[Disjoint sets|disjoint]] for sets).
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| If ''A'' and ''B'' are disjoint events, then ''P''(''A''∪''B'') = ''P''(''A'') + ''P''(''B''). This extends to a (finite or countably infinite) sequence of events. However, the probability of the union of an uncountable set of events is not the sum of their probabilities. For example, if ''Z'' is a [[normal distribution|normally distributed]] random variable, then ''P''(''Z''=''x'') is 0 for any ''x'', but ''P''(''Z''∈'''R''') = 1.
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| The event ''A''∩''B'' is referred to as “''A'' and ''B''”, and the event ''A''∪''B'' as “''A'' or ''B''”.
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| ==See also==
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| * [[Sigma-algebra]]
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| * [[Space (mathematics)]]
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| * [[Measure space]]
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| * [[Fuzzy measure theory]]
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| * [[Filtered probability space]]
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| * [[Talagrand's concentration inequality]]
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| ==Bibliography==
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| * [[Pierre Simon de Laplace]] (1812) ''Analytical Theory of Probability''
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| :: The first major treatise blending calculus with probability theory, originally in French: ''Théorie Analytique des Probabilités''.
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| *[[Andrei Nikolajevich Kolmogorov]] (1950) ''Foundations of the Theory of Probability''
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| :: The modern measure-theoretic foundation of probability theory; the original German version (''Grundbegriffe der Wahrscheinlichkeitrechnung'') appeared in 1933.
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| * [[Harold Jeffreys]] (1939) ''The Theory of Probability''
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| :: An empiricist, Bayesian approach to the foundations of probability theory.
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| * [[Edward Nelson]] (1987) ''Radically Elementary Probability Theory''
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| :: Discrete foundations of probability theory, based on nonstandard analysis and internal set theory. downloadable. http://www.math.princeton.edu/~nelson/books.html
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| * [[Patrick Billingsley]]: ''Probability and Measure'', John Wiley and Sons, New York, Toronto, London, 1979.
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| * Henk Tijms (2004) ''Understanding Probability ''
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| :: A lively introduction to probability theory for the beginner, Cambridge Univ. Press.
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| * David Williams (1991) ''Probability with martingales''
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| :: An undergraduate introduction to measure-theoretic probability, Cambridge Univ. Press.
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| *{{cite book
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| | last = Gut
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| | first = Allan
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| | title = Probability: A Graduate Course
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| | publisher = Springer
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| | year = 2005
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| | isbn = 0-387-22833-0
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| }}
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| ==External links==
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| * {{Springer |title=Probability space |id=P/p074960 |first=V.V. |last=Sazonov}}
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| * [http://www.youtube.com/watch?v=9eaOxgT5ys0 Animation] demonstrating probability space of dice
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| * [http://www.math.uah.edu/stat/ Virtual Laboratories in Probability and Statistics] (principal author Kyle Siegrist), especially, [http://www.math.uah.edu/stat/prob Probability Spaces]
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| * [http://en.citizendium.org/wiki/Probability_space Citizendium]
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| *[http://www.encyclopediaofmath.org/index.php/Probability_space Complete probability space]
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| * {{MathWorld |urlname=ProbabilitySpace}}
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| [[Category:Probability theory]]
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