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?
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$E_1, E_2, E_3$ are three jointly independent events. The, the events $E_1 \cup E_2$ and $E_3$ are pairwise independent.
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$E_1, E_2, E_3$ 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 $P(AB)=P(A)P(B)$.
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True. Jointly independent events are always pairwise independent.
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I am unsure about how to tackle this one.
Best Answer
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.
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.