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.
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.
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.