Theory 1
Two events are independent when information about one of them does not change our probability estimate for the other. Mathematically, there are three ways to express this fact:
Independence
Events
and are independent when these (logically equivalent) equations hold:
The last equation is symmetric in
and .
- Check:
and - This symmetric version is the preferred definition of the concept.
Multiple-independence
A collection of events
is mutually independent when every subcollection satisfies: A potentially weaker condition for a collection
is called pairwise independence, which holds when all 2-member subcollections are independent: One could also define
-member independence, or -member independence. Plain ‘independence’ means any-member independence.