Independence (probability theory)

When dealing with collections of more than two events, there are two different notions of independence. A finite set of events \{A_i\}_{i=1}^nis,
* Pairwise independence: if every pair of events is independent, P[A_j \cap A_k] = P[A_j] P[A_k]
* Mutual independence: if every event is independent of the intersection of all subsets of the set of remaining events, that is iff:
** For any event A_i, we have P[\bigcap_{j=1}^k A_{i_j}] = \prod_{j=1}^k P[A_{i_j}] for every k \leq n and for every subset of this size with indices 1 \leq i_1 < \cdots < i_k \leq n
** Or equivalently but without possibly confusing indices: For any event A_i, we have P[A_i \cap (\bigcap_{j=1}^k A_j)] = P[A_i] \prod_{j=1}^k P[A_j] for any 1 \leq j < k \leq n. (NOTE: DO WE HAVE TO EXCLUDE A_i ITSELF FROM HERE?)
When people refer to independence without being specific, this generally means mutual independence.
== What does it look like on a Venn-diagram? ==
See math notes #8 intermezzo. This video was useful/could be embedded on this page.