Questions about the independence of three events I have several statements related to the probability of three events and am trying to determine if the statements are true or false.  For some of the answers, I have intuition but I would prefer to learn a provable answer. 


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*I have 3 events $E_1, E_2, E_3$ in a probability space.  If $E_3 \subset E_1$ and $E_1, E_2$ are pairwise independent, are $E_2, E_3$ pairwise independent?

*$E_1, E_2, E_3$ are three jointly independent events. The, the events $E_1 \cup E_2$ and $E_3$ are pairwise independent.

*$E_1, E_2, E_3$ be three pairwise independent events in a probability space.  Then the events $E_1\cup E_2$ and $E_3$ are pairwise independent.
My solutions thus far.


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*My thought is it depends on the amount of space $E_3$ occupies of $E_1$ because independence is defined as $P(AB)=P(A)P(B)$.

*True. Jointly independent events are always pairwise independent. 

*I am unsure about how to tackle this one.
 A: For (1), pick $E_2,E_3$ your favorite dependent events and let $E_1$ be the sample space itself, the sure event.  $E_1$ trivially contains $E_3$ and further $E_1$ is trivially independent with any other event.
For (2), this is a fair bit of algebra but fairly straightforward to prove.  Just remember how intersection distributes over union, apply inclusion-exclusion and properties of independence, and factor.

 $P((E_1\cup E_2)\cap E_3) = P((E_1\cap E_3)\cup (E_2\cap E_3))$
$ = P(E_1\cap E_3) + P(E_2\cap E_3) - P(E_1\cap E_2\cap E_3) $
$= P(E_1)P(E_3) + P(E_2)P(E_3) - P(E_1)P(E_2)P(E_3)$
$=P(E_3)\times (\cdots)$

For (3), consider an example where the events are pairwise but not mutually independent.  For example considering the uniform distribution over the sample space $\{1,2,3,4\}$ and letting the event $E_i = \{i,4\}$.  You have $E_1,E_2,E_3$ are each pairwise independent but not mutually independent.

 Here we have $P(E_1\cup E_2)=\frac{3}{4}$, $P(E_3)=\frac{1}{2}$, and $P((E_1\cup E_2)\cap E_3) = \frac{1}{4}\neq \frac{3}{4}\times\frac{1}{2}$

