If event $A$ happens with probability $p$, then when is event $B$ guaranteed to not be mutually exclusive with $A$? Say event $A$ happens with probability $p$. We want to find the probability $q$ of $B$ happening such that it ensures that $A$ and $B$ are not mutually exclusive. Is this simply $q > 1-p$?
We have the relation
$$
P(A \cup B) = P(A) + P(B) - P(A \cap B)
$$
If $A$ and $B$ are mutually exclusive then $P(A \cap B) = 0$
If $q > 1-p$, then P(A)+P(B) > 1, so this implies they're not mutually exclusive otherwise P(A \cup B) > 1

Yes, that's exactly the condition that does it.  And you can't do better than that because if $B = A^c$, they are mutually exclusive with $p+q=1$.  
