Editing Probability space (section)

==== 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>.