$A, B, C$ are mutually independent. What is the $P(AB\cup C)$?
My guess
$$P(AB \cup C)=P(AB)+P(C)=P(A)P(B)+P(C) $$ Is this correct or do I have to add some subtraction [$-P(A\cap B \cap C)$]? I guess, I don't have a full understanding of independent events.
You have to subtract $P(A \cap B \cap C)$ since the term need not be zero.
\begin{align} P(AB \cup C)&=P(AB)+P(C)-P(ABC)\\&=P(A)P(B)+P(C)-P(A)P(B)P(C) \end{align}
A particular examples of $A,B,C$ that are mutually independent and are not mutually exclusive are whether you get heads from 3 independent coin tosses.