Questions regarding mutual independence of events

Question

Have a few questions regarding mutual independence:

1. If I have a set of events $A_1, A_2, …A_n$ that are all pairwise independent, it is possible that the events may not be mutually independent?

2. If I have n events and $P(A_1 \cap A_2 \cap … A_n) = P(A_1)P(A_2)…P(A_n)$, it is possible that the events are not pairwise independent (and hence not mutually independent)?

3. If I have a sample space $ \Omega = A_1 \cup A_2 \cup … A_n $ where $A_1, A_2, …A_n $ are all pairwise disjoint, does this mean that $A_1, A_2, …A_n $ are also mutually independent (not just pairwise independent)?

Answer 1

1. Yes it is possible. See https://en.wikipedia.org/wiki/Pairwise_independence for a standard example.

2. Yes it is possible, let $A_1 = \emptyset$ and the statement holds regardless of the rest of the $A_i$, which do not have to be independent.

3. No this does not imply mutual independence, or even pairwise independence. If $A_1$ and $A_2$ are disjoint, they cannot be independent unless at least one of them has probability zero. Intuitively, if I told you that $A_1$ occurred, then you know for certain that $A_2$ did not occur.