Questions regarding mutual independence of events [closed]

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)?

closed as off-topic by Kavi Rama Murthy, max_zorn, Cesareo, Javi, José Carlos SantosApr 18 at 10:42

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1 Answer

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.