Editing Probability space (section)

== 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: <math>\Omega = \{\text{H}, \text{T}\}</math>. The σ-algebra <math>\mathcal{F} = 2^{\Omega}</math> contains <math>2^2 = 4</math> events, namely: <math>\{\text{H}\}</math> ("heads"), <math>\{\text{T}\}</math> ("tails"), <math>\{\}</math> ("neither heads nor tails"), and <math>\{\text{H}, \text{T}\}</math> ("either heads or tails"); in other words, <math>\mathcal{F} = \{\{\}, \{\text{H}\}, \{\text{T}\}, \{\text{H}, \text{T}\}\}</math>. There is a fifty percent chance of tossing heads and fifty percent for tails, so the probability measure in this example is <math>P(\{\}) = 0</math>, <math>P(\{\text{H}\}) = 0.5</math>, <math>P(\{\text{T}\}) = 0.5</math>, <math>P(\{\text{H}, \text{T}\}) = 1</math>.

==== Example 2 ====
The fair coin is tossed three times. There are 8 possible outcomes: {{math|1=Ω = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}<nowiki/>}} (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>\mathcal{F} = 2^\Omega</math> of {{math|1=2<sup>8</sup> = 256}} 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 {{math|1=Ω = ''A''<sub>1</sub> ⊔ ''A''<sub>2</sub> = {HHH, HHT, THH, THT} ⊔ {HTH, HTT, TTH, TTT}<nowiki/>}}, where ⊔ is the ''[[disjoint union]]'', and the corresponding σ-algebra <math> \mathcal{F}_\text{Alice} = \{\{\}, A_1, A_2, \Omega\}</math>. Bryan knows only the total number of tails. His partition contains four parts: {{math|1=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}<nowiki/>}}; accordingly, his σ-algebra <math> \mathcal{F}_\text{Bryan}</math> contains 2<sup>4</sup> = 16 events.

The two σ-algebras are [[comparability|incomparable]]: neither <math> \mathcal{F}_\text{Alice} \subseteq \mathcal{F}_\text{Bryan}</math> nor <math> \mathcal{F}_\text{Bryan} \subseteq \mathcal{F}_\text{Alice}</math>; both are sub-σ-algebras of 2<sup>Ω</sup>.

==== 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 [[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, Bryan) is different from (Bryan, Alice). 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 <math> \mathcal{F}_\text{Alice}</math> 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 ∅.

Bryan knows the exact number of voters who are going to vote for Schwarzenegger. His incomplete information is described by the corresponding partition {{math|1=Ω = ''B''<sub>0</sub> ⊔ ''B''<sub>1</sub> ⊔ ⋯ ⊔ ''B''<sub>100</sub>}} and the σ-algebra <math> \mathcal{F}_\text{Bryan}</math> consists of 2<sup>101</sup> events.

In this case, Alice's σ-algebra is a subset of Bryan's: <math> \mathcal{F}_\text{Alice} \subset \mathcal{F}_\text{Bryan}</math>. Bryan's σ-algebra is in turn a subset of the much larger "complete information" σ-algebra 2<sup>Ω</sup> consisting of {{math|2<sup>''n''(''n''−1)⋯(''n''−99)</sup>}} events, where ''n'' is the number of all potential voters in California.

=== Non-atomic examples ===

==== Example 4 ====
A number between 0 and 1 is chosen at random, uniformly. Here Ω = [0,1], <math> \mathcal{F}</math> is the σ-algebra of [[Borel set]]s on Ω, and ''P'' is the [[Lebesgue measure]] on [0,1].

In this case, the open intervals of the form {{open-open|''a'',''b''}}, where {{math|0 < ''a'' < ''b'' < 1}}, could be taken as the generator sets. Each such set can be ascribed the probability of {{math|1=''P''((''a'',''b'')) = (''b'' − ''a'')}}, which generates the [[Lebesgue measure]] on [0,1], and the [[Borel σ-algebra]] on Ω.

==== Example 5 ====
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 {{math|1={(''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>}<nowiki/>}} 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 {{math|(''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>.

These two non-atomic examples are closely related: a sequence {{math|(''x''<sub>1</sub>, ''x''<sub>2</sub>, ...) ∈ {0,1}<sup>∞</sup>}} leads to the number {{math|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-pathological 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.