Editing Conditional independence (section)

===Examples===

==== 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>

==== Proximity and delays ====
Let events A and B be defined as the probability that person A and person B will be home in time for dinner where both people are randomly sampled from the entire world. Events A and B can be assumed to be independent i.e. knowledge that A is late has minimal to no change on the probability that B will be late. However, if a third event is introduced, person A and person B live in the same neighborhood, the two events are now considered not conditionally independent. Traffic conditions and weather-related events that might delay person A, might delay person B as well. Given the third event and knowledge that person A was late, the probability that person B will be late does meaningfully change.<ref name=":0">[https://math.stackexchange.com/q/23093 Could someone explain conditional independence?]</ref>

==== Dice rolling ====
Conditional independence depends on the nature of the third event. If you roll two dice, one may assume that the two dice behave independently of each other. Looking at the results of one die will not tell you about the result of the second die. (That is, the two dice are independent.) If, however, the 1st die's result is a 3, and someone tells you about a third event - that the sum of the two results is even - then this extra unit of information restricts the options for the 2nd result to an odd number. In other words, two events can be independent, but NOT conditionally independent.<ref name=":0" />

====Height and vocabulary====
Height and vocabulary are dependent since very small people tend to be children, known for their more basic vocabularies. But knowing that two people are 19 years old (i.e., conditional on age) there is no reason to think that one person's vocabulary is larger if we are told that they are taller.