# 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

## 1 Answer

Events are sets of outcomes in a sample space. Probability is a measure of these sets.

Two events are mutually exclusive if they have no common outcome.   Therefore the measure of their union is the sum of their measure.   $A\cap B=\emptyset \implies \mathsf P(A\cup B)=\mathsf P(A)+\mathsf P(B)$

For your intutition: I pour oil and water into an empty container and, because the oil and water are immissable, therefore the volume in the container equals the sum of the volumes of each.

• 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
• Cool. I had not understood it because I was reading in my phone and it didn't display the Greek letters. Ok, I got your words very well. Thank you. Um, I got another dumb question. If we assumed a heart-attack and brain death are not related, and we had the probability of dying from a heart-attack and the prob. from dying of brain death, (here´s the dumb part haha), the body or the circulation system or whatever link they have in common is the sample space? If so, um, dying from both failures would be an example of using the addition rule? Thanks for your time too, as Vaas. – snzcc Oct 4 '17 at 1:38
• @Snzcc No, not remotely. You are confusing independence and mutual exclusion; they are not the same at all. You are also confusing union and intersection. Independent events will have common outcomes, and the set of such is the intersection of the events. The probability for the intersection of independent events equals the product of the probabilities for each event. – Graham Kemp Oct 4 '17 at 1:50
• @Snzcc PS: The sample space is the outcome set, the set of measurable events, and the probability measure. In this case, the outcome set would be the set of causes of death. – Graham Kemp Oct 4 '17 at 1:57