For my probability class, we were asked to solve a few questions about the Principle of Inclusion-Exclusion:
Prove they are true, or provide a counter example.
1) If P(A)+P(B)+P(C) = 1, are A,B,C mutually exclusive?
2) If $P(A \cup B \cup C) = 1$, are these events mutually exclusive?
So, I was trying to attempt the contrapositive for both since it seemed easier. To assume (for either problem) that they are mutually exclusive would be to say that $A \cap B = \emptyset$. We can also say that $B\cap C = \emptyset$, and again for $A \cap C$. Lastly, we know that $A\cap B\cap C = \emptyset$.
This means that (based on the principle of inclusion-exclusion), $P(A\cup B\cup C) = P(A)+P(B)+P(C)$.
I don't know where to go from here; I know all event-intersections are empty but am not sure how to proceed.
If you're trying to prove the contrapositive, you need to start by assuming they are not mutually exclusive.