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I'm having trouble understanding this statement: "any two deterministic quantities are independent"

The example the text provides is as follows: $$Prob(\varnothing\cap\Omega) = Prob(\varnothing) = 0 = Prob(\varnothing)P(\Omega) $$ Which proves $\varnothing$ and $\Omega$ are independent.

However, we cannot say that any event A and its complement, $A^C$, is necessarily independent(at least I don't think). My question is why are two deterministic quantities are independent? (Maybe I'm just not understanding what deterministic in this context means?)

Any guidance is greatly appreciated :)

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    $\begingroup$ I would guess that "deterministic" means the probability is either $0$ or $1$. (It's bound to happen or it will never happen.) In which case, two deterministic events are clearly independent. $\endgroup$ – saulspatz Dec 10 '18 at 16:53
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    $\begingroup$ @WilsonGuo Related to the saulspatz comment, can you prove the following? Let $A$ be an event with $P[A]=1$. Let $B$ be any other event. Use the definition of independent to prove that $A$ and $B$ are independent. [Hint: $B= (B\cap A) \cup (B \cap A^c)$. Or perhaps just prove $A^c$ and $B$ are independent.] $\endgroup$ – Michael Dec 10 '18 at 17:02

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