So in my intro course to stats, we ecnountered the law of total probability. The definition is $$P(A) = \sum^n_{j=1}P(A\mid H_j)P(H_j)$$ However, the definition says
$$\bigcup^n_{j=1}H_j = S \quad \text{(Union of all Events form Sample Space)}\\ \bigcap^n_{j=1}H_j=\emptyset \quad \text{Events are Pairwise Disjoint}$$
Now I was wondering what happens, when they tell me (in the exam) to calculate the total probability of an event $A$, even though an intersection of H_1 and H_2 exist.
So I constructed the following example and wanted to ask for your help on whether that is correct.
So I constructed two Events $A,B$ with $P(A) = 0.5, P(B) = 0.7 \text{ and } P(A \cap B) = 0.2$
I then introduce an event $M,$ which can happen according to the following probabilites. $$P(M\mid A \setminus B)=P(M\mid A \cap B)=P(M\mid B \setminus A) = \frac{1}{3}$$
This introduces the next drawing.
And thus I ultimately calculated $$P(M) = P(A \setminus B\mid M) P(A \setminus B) +P(A \cap B \mid M)P(A \cap B) + P(B \setminus A \mid M) P(B \setminus A) = \frac{1}{3}\cdot(0.3+0.2+0.5)= \frac{1}{3}$$
Is that correct? Is thus the tric to redefine all the events of the subspace as pairwise disjoint? Or is there maybe another way/trick or something one should pay attention to?