# 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.

1. 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?

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

3. $$E_1, E_2, E_3$$ 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.

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

2. True. Jointly independent events are always pairwise independent.

3. I am unsure about how to tackle this one.

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.
$$P((E_1\cup E_2)\cap E_3) = P((E_1\cap E_3)\cup (E_2\cap E_3))$$ $$= P(E_1\cap E_3) + P(E_2\cap E_3) - P(E_1\cap E_2\cap E_3)$$ $$= P(E_1)P(E_3) + P(E_2)P(E_3) - P(E_1)P(E_2)P(E_3)$$ $$=P(E_3)\times (\cdots)$$
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 $$P((E_1\cup E_2)\cap E_3) = \frac{1}{4}\neq \frac{3}{4}\times\frac{1}{2}$$