# Is the logic of addition and multiplication for probabilities related to something else?

I mean, is there a mathematical argument or best said, relationship, to another field of math, the nature of adding or multiplicating two probabilities? Like, could we think that we add things, imagining we didn't "reason" or made it intuitively because they are exclusive? I know this may seem senseless because two exclusive quantities, as far as I get it, wouldn't add at all (I imagined water and oil; also if I got stupid ideas or I'm confusing things, let me know, please)

Sorry if this question is dumb, I have only taken one class of probability and stats and it was more of the applications of these topics rather than discussing them. It would be cool if y'all also could recommend me a book about "math discussion" for newbies or ideas to develop a more mature appreciation of math topics, rather than only getting the idea and knowing how to use the concept as we do in class. I recently started rediscovering how fantastic maths and logic are after reading a couple of discussions here.

Thanks for anyone who spends his time answering!

• adding water and oil creates a mixture. you said you've taken a class in probability and stats have they defined what a "probability" is? – Vaas Oct 4 '17 at 0:42
• Ok, I got it wrong. But what could be an example to try to explain the idea of addition/multiplication I tried to point out? – snzcc Oct 4 '17 at 0:48
• sorry didnt mean to come across pedantic, can you give a very brief overview of what you've learnt in your class? or at least what sort of level youre at, (Uk examples. GCSE, A-level, Undergrad etc)...just so i have an idea of where youre at. – Vaas Oct 4 '17 at 0:51
• Undergrad (yes, I''m sorry of asking such a dumb question at this level). Uh, we basically covered probability in general terms (uh, probability distributions and all of that). I don't remember now the topics, but sure thing I can get the idea or search the basis of the idea later. – snzcc Oct 4 '17 at 0:52
• not at all stupid, but if you were at say gcse the answer would have to be tailored differently – Vaas Oct 4 '17 at 0:54

Two events are mutually exclusive if they have no common outcome.   Therefore the measure of their union is the sum of their measure.   $A\cap B=\emptyset \implies \mathsf P(A\cup B)=\mathsf P(A)+\mathsf P(B)$
• to add to graham kemp's answer, you can consider a single set (denoted $\Omega$) of all possible elementary outcomes(denoted $\omega$) to be the sample space then the event A (as said above) is a subset of the sample space. and you say A occurs if $\omega \in A$. P is a function that maps the subset to [0,1] (if i remember rightly) on a personal note. Graham Kemp; you state probability is a measure of these sets. can we consider the probability function to be a metric on the sample space? – Vaas Oct 4 '17 at 1:01
• @Vaas You remember rightly. Also $(1) \mathsf P(\emptyset)=0, (2)\mathsf P(\Omega)=1,$ and $(3)$ for countable disjoint subsets $(C_i)$ of $\Omega$, then $P(\bigcup_i C_i)=\sum_i \mathsf P(C_i)$ – Graham Kemp Oct 4 '17 at 1:06