Conditional independence

Parent: Probability theory Source:  http://en.wikipedia.org/wiki/Conditional_independence

$A$ and $B$ are conditionally independent given $C$ $\Leftrightarrow$ given the knowledge that $C$ occurs, the knowledge of whether $A$ occurs provides no information whatsoever on the likelihood of $B$ occurring, and vice versa.

Examples

Weather and delay

• Let the two events be the probabilities of persons $A$ and $B$ getting home in time for dinner
• The third event $C$ is the fact that a snow storm hit the city.
• While both $A$ and $B$ have a lower probability of getting home in time for dinner, the lower probabilities will still be independent of each other. i.e. the knowledge that $A$ is late does not tell you whether $B$ will be late.
• However, if you have information (other than $C$) that they live in the same neighborhood, use the same transportation, and work at the same place, then the two events are NOT conditionally independent.

Height and vocabulary

Height and vocabulary are independent; but they are conditionally not independent if you add age.