141. Independence
Two events are said to be independent if the occurrence of one event does not change our beliefs about the other
Events A and B are independent from each other if:
Implies:
Applies even if
141.1. Independence of Complements
If A and B are independent, then the following pairs are also independent:
- and
- and
- and
Formally:
If
then:
Proof
Start with:
Since and are disjoint, by the Additivity Axiom:
Rearanging:
Thus we are back at out definition of independence:
141.2. Conditional Independence
141.3. Pairwise Independence
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Mutual (collective) independence pairwise independence
If a collection of events is mutually independent, then every pair of events in the collection is independent.
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Pairwise independence mutual (collective) independence
A collection of events can be pairwise independent but not mutually independent.