What is the probability that this happen P(A)+P(B)−2P(A ∩B) I think that my question has a bad structured, but my question is in base the next.
Let A and B be any sets. Show that the probability that exactly
one of the events A or B occurs is:
$$P(A)+P(B)−2P(A\cap B)$$
I thought that this is possible with The inclusion-exclusion principle, is it right or I need other thing?

The inclusion exclusion principle (for any two events, $A,B$) is that:$$\mathsf P(A\cup B)=\mathsf P(A)+\mathsf P(B)-\mathsf P(A\cap B)\\~\\\text{also}\\~\\\mathsf P(A\cap B)=\mathsf P(A)+\mathsf P(B)-\mathsf P(A\cup B)$$
You may indeed use this to show that: $$\mathsf P((A\cup B)\cap (A\cap B)^\complement)=\mathsf P(A)+\mathsf P(B)-2\,\mathsf P(A\cap B)$$

Also note:
$(A\cup B)\cap (A\cap B)^\complement= (A\cap B^\complement)\cup(A^\complement\cap B)$
