Editing Conditional independence (section)

==== Coloured boxes ====
Each cell represents a possible outcome. The events <math>\color{red}R</math>, <math>\color{blue}B</math> and <math>\color{gold}Y</math> are represented by the areas shaded {{font color|red|red}}, {{font color|blue|blue}} and {{font color|gold|yellow}} respectively. The overlap between the events <math>\color{red}R</math> and <math>\color{blue}B</math> is shaded   {{font color|purple|purple}}.

[[Image:Conditional independence.svg|450px|These are two examples illustrating '''conditional independence'''.]]

The probabilities of these events are shaded areas with respect to the total area. In both examples <math>\color{red}R</math> and <math>\color{blue}B</math> are conditionally independent given <math>\color{gold}Y</math> because:

:<math>\Pr({\color{red}R}, {\color{blue}B} \mid {\color{gold}Y}) =  \Pr({\color{red}R} \mid {\color{gold}Y})\Pr({\color{blue}B} \mid {\color{gold}Y})</math><ref>To see that this is the case, one needs to realise that Pr(''R'' ∩ ''B'' | ''Y'') is the probability of an overlap of ''R'' and ''B'' (the purple shaded area) in the ''Y'' area. Since, in the picture on the left, there are two squares where ''R'' and ''B'' overlap within the ''Y'' area, and the ''Y'' area has twelve squares, Pr(''R'' ∩ ''B'' | ''Y'') = {{sfrac|2|12}} = {{sfrac|1|6}}. Similarly, Pr(''R'' | ''Y'') = {{sfrac|4|12}} = {{sfrac|1|3}} and Pr(''B'' | ''Y'') = {{sfrac|6|12}} = {{sfrac|1|2}}.</ref>

but not conditionally independent given <math>\left[ \text{not }{\color{gold}Y}\right]</math> because:

:<math>\Pr({\color{red}R}, {\color{blue}B} \mid \text{not } {\color{gold}Y}) \not= \Pr({\color{red}R} \mid \text{not } {\color{gold}Y})\Pr({\color{blue}B} \mid \text{not } {\color{gold}Y})</math>