Editing Probability space (section)

==== 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.