Caution: I modified this original answer to simplify the examples.
You add probabilities when the events you are thinking about are alternatives
(eg: A soccer team scores 0 goals or 1 goal or 2 goals in their match).
You are looking for "mutually exclusive" events: things which could NOT happen at the same time (in the same match).You multiply probabilities when you wish two or more different things to happen "at the same time" or "consecutively" (eg: England scores 1 and Scotland scores 1 and Wales scores 2).
The key thing here is that the events are independent: they do not affect each other, or the second does not affect the first (etc).
How can the question entitled above, be explained and resolved even more intuitively than the following? Please do not answer with formal proofs; I pursue only intuition.
I already understand, and so ask NOT about, the following which relies on algebra and so reveals no intuition:
For mutually exclusive events, $\color{forestgreen}{Pr(A \cap B) = 0} \implies Pr(A \cup B) = Pr(A) + Pr(B) – \color{forestgreen}{0}$.
For independent events, $\color{forestgreen}{Pr(A|B) = Pr(A)} \implies Pr(A \cap B) = Pr(A|B) \times Pr(B) = \color{forestgreen}{Pr(A)} \times Pr(B).$
Best Answer
Sequential events. Event A happens $P_a$ proportion of the time. Then B happens $P_b$ porportion of the time. So Event B happen after evant A will happen $P_b$ proportion of the $P_a$ proportion of the time.
Mutually exclusive events: Event A happens $P_a$ proportion of the time. That's $P_at$ total occurrence of some $t$ representing time. Event B is completely independent. It occurs $P_bt$ total. There is not overlap. So the totality of the time when one or the other occurred it $P_at$ + $P_bt$. Probability is $P_a + P_b$.