# Meaning of “Any two deterministic quantities are independent”

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 :)

• 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. – saulspatz Dec 10 '18 at 16:53
• @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.] – Michael Dec 10 '18 at 17:02