# How can we define something and have it translate to the real world?

I'm looking at a definition in my textbook (although my question applies to most definitions)

Let A, B be two events. Define $P(B|A)$ i.e. the probability of B given A, by: $$P(B|A) = \frac{P (A\cap B) }{P(A)}$$

How can we define $P(B|A)$ in this way? Shouldn't it be a theorem? How can I calculate the probability of an event B in the real world given an event A in the real world, with this definition?

How can we define something and then use it to answer quesitons about the real world?

Another example: Where is the proof that $$\iiint_K1dxdydx$$ where $K$ is an abstract sphere with radius $r$, actually calculates the volume of such a sphere in the real world?

1. How can we define something and have it translate to the real world?
2. Why is it that mathematical models translates to the real world at all?
• These are philosophical, not mathematical, questions. – Lord Shark the Unknown Aug 29 '17 at 18:13
• The latter is true because the volume of a set is defined as (three-dimensional) Lebesgue measure of that set and this happens to be the same as said integral over the given set. There is nothing more that can be said here and I don't actually think this definition really helps you personally if you have not studied measure theory yet. - But you might as well take the above integral as the definition of volume. So yeah, mostly a philosophical question. – Stefan Perko Aug 29 '17 at 18:17
• "Shouldn't it be a theorem?"... without a definition of something, how do you think of a theorem related to that? – MAN-MADE Aug 29 '17 at 18:24
• @MANMAID Of course $P(B|A)$ is a definition. But how can we define that $P(B|A)$ is equal to $\frac{P(A\cup B)}{P(A)}$ – Heuristics Aug 29 '17 at 18:39
• @Heuristics have you read independence? Note that $P(A\cap B)=P(B\mid A)P(A)$. Btw $P(A)>0$ – MAN-MADE Aug 29 '17 at 18:41

First of all, it should be $$P(B|A) = \frac{P (A\cap B) }{P(A)},$$ and then, that wasn't some arbitrary definition, but a generalization of simple models in finite probability spaces. And then, it's a somewhat unfortunate example: nothing is more controversial than the real world interpretation of probabilities, especially conditional probabilities (and Bayesian statistics is yet another can of worms). Concerning that sphere: it's an abstract sphere. Does it exist in the real world? No, not exactly, look at some almost ideally polished metallic sphere through an electron microscope. It looks a bit rough and cubistic, almost like the integral sums we use to define integrals, doesn't it?