I had a question regarding the properties of probability.
If we were to say that we have "any two events $A$ and $B$," is it okay to assume that they both belong to the same sample space $S$ (i.e. $A \subseteq S$ and $B \subseteq S$)?
I'll give a specific example exercise that prompted me to ask this question:
"Show that for any events $A$ and $B$, $\ $ $P(A) + P(B) - 1 \le P(A \cap B)$."
What I did is move the 1 over, so now we have
$$P(A) + P(B) \le 1 + P(A \cap B)$$
which is true if we assume that $P(A) + P(B) \le P(S) = 1$.
I hope my question makes sense. Thank you for the feedback!