Addition rule

The general addition rule is $P(A \cup B) = P(A) + P(B) - P(A \cap B)$, where $\cup$ is the union sign, read "A or B", and $\cap$ is the intersection sign, read "A and B". It holds for every pair of events, with nothing to check first. The subtraction is there because any outcome living in both events was counted once inside $P(A)$ and a second time inside $P(B)$, so it has to come back out once. "Or" is inclusive: $A \cup B$ means at least one of the two, including the outcomes where both happen.

Draw one card from a standard 52-card deck. There are 13 hearts and 4 kings, and exactly one card, the king of hearts, is both. So $P(\text{heart or king}) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} \approx 0.3077$. Listing them agrees: 13 hearts plus the 3 kings that are not hearts is 16.

"The two events are different, so add them: $13/52 + 4/52 = 17/52$." That counts the king of hearts twice, and the list has only 16 cards in it. Plain addition is legal only when $P(A \cap B) = 0$, which is what mutually exclusive events give you: they share no outcome at all, so the subtracted term was zero anyway. The plain sum is a consequence of the general rule, never the rule itself.

For three events the pattern extends: add the three single probabilities, subtract the three pairwise intersections, then add the triple intersection back. Roll one fair die with $A = \{2, 4, 6\}$, $B = \{4, 5, 6\}$ and $C = \{2, 3, 5\}$, so the pairs hold 2, 1 and 1 outcomes and no outcome is in all three. The rule gives $\frac{3 + 3 + 3 - 2 - 1 - 1 + 0}{6} = \frac{5}{6} \approx 0.8333$, and the union is $\{2, 3, 4, 5, 6\}$, five outcomes. Reverse those signs, adding the pairs and subtracting the triple, and you get $13/6 \approx 2.17$. A probability above 1 is the tell.

Unions of events are topic 2.7.