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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, $\color{forestgreen}{Pr(A|B) = Pr(A)} \implies Pr(A \cap B) = Pr(A|B) \times Pr(B) = \color{forestgreen}{Pr(A)} \times Pr(B).$

Community
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  • One source of intuition is geometric (aka measure-theoretic). The area of two nonoverlapping sets ought to be the sum of their areas. The area of an overlap of two sets A and B ought to be the area of A times the fraction of the area of A that the overlap takes up (or similarly for B). Think of throwing darts at certain regions on a dartboard. – symplectomorphic Feb 02 '16 at 06:57
  • @symplectomorphic I got the area part but can you please explain how this carries on to finding the probability of the intersection of two events? – HeWhoMustBeNamed Jan 01 '18 at 12:20

2 Answers2

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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 $P_a + P_b$.

fleablood
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  • Where in the world did I ever say anything about the order of the events? There's nothing here about which event happened first. Just whether they both happened. – fleablood Dec 31 '17 at 18:43
  • So I did. I wrote this almost two freakin' years ago so I can barely remember. – fleablood Jan 01 '18 at 04:11
  • So can you address my doubt: From what you wrote, I got that in order to calculate the probability of A and B happening we have to multiply P(A) and P(B). But aren't there 2 possibilities: that of A happening after B and that of B happening after A? So, why isn't the total probability twice of what you calculated? – HeWhoMustBeNamed Jan 01 '18 at 07:00
  • "So can you address my doubt?" No. I can not. I wrote it two years ago in response to a question that was altered and changed a year after I answered it. I have no memory what the original question was or why I answered what I did. but it was meant to be an intuitive, not rigorous, explanation of why we multiply probabilities if two events happen but add if one or the other happen. I'm not sure what the stuff about time was for. – fleablood Jan 01 '18 at 07:28
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When it comes to addition I'd take the integral theoretic approach. You just add the events and the sum of the probabilities are the expected value of the sum of the outcomes. For mutually exclusive events it will never happen that both events happen, for non-exclusive events the sum of the outcomes is one if only one event happens and two if both (and of course zero if none).

Multiplication on the other hand requires them to be independent for it to make sense intuitively, to me at least. I still have the integral theoretic approach and the product of two integrals does not seem to have any relation to a single integral unless the events are independent.

skyking
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