Can we assume that “any two events” are within the same sample space?

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 $B$, $ $ $P(A) + P(B) – 1 le P(A cup B)$.”**

What I did is move the one over, so now we have

$$P(A) + P(B) le 1 + P(A cup B)$$

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!

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