Why is this particular case often used to introduce the property $P(A \cup B) = P(A) + P(B) - P(A \cap B)$? I'm taking up a probability course and my teacher, when explaining the fondamental properties of probability, listed this one:
$$\text{If } A \text{ and }B \text{ are mutually exclusive events, then }P(A \cup B)  = P(A) + P(B)$$
Then, later, under an &quot;other properties&quot; list, you had:
$$P(A \cup B)  = P(A) + P(B) - P(A \cap B)$$
which, per my understanding, is the general case of the first statement.
I have seen this approach taken in some books and by other teachers as well, where the particular case where the probability of the intersection is $0$ is treated as the &quot;main&quot; case, and the general case is listed separately, as if the two had no correlation, whereas the former is really just the latter with the additional hypothesis that $A \cap B = \emptyset$.
Is there any reason why this is? Are there any cases in which, mathematically, it makes more sense to treat a particular case as the main instance of a property or theorem?
We have $A = (A \smallsetminus B) \cup (A \cap B)$ and $B = (B \smallsetminus A) \cup (A \cap B)$. Notice that $A \smallsetminus B$ and $A \cap B$ are disjoint. Similarly $B \smallsetminus A$ and $A \cap B$ are disjoint. Now
$$ A \cup B = (A \smallsetminus B) \cup (A \cap B) \cup (B \smallsetminus A) \text{.}$$ The sets in the previous equation are pairwise disjoint. Use the equations for the probabilities of disjoint sets and some algebra to calculate the probability of $A \cup B$.
This is best viewed by drawing some venn diagrams.
In the first case, if it is said that they are mutually exclusive (never touching or intersecting) then we can simply add up the elements in A and B. The venn diagram for this would look like two circles labeled A and B separated from each other.
Otherwise, if nothing is said about being mutually exclusive, we assume that they are touching or intersecting and so you need to subtract or take away the overlapping part as to not double count the middle area (because we already include all of A and all of B). The venn diagram for this would be two circles labeled A and B with some overlap in the middle.
The two instances mean two different things and they usually start by teaching the nonoverlap property before learning about the other one.