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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?
Heuristics
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    These are philosophical, not mathematical, questions. – Angina Seng 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
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    "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
  • We can define things in any way we want, and we apply it to the real world when we believe the description of the definition applies. – Bram28 Aug 29 '17 at 18:42
  • See Conditional probability for an explanation. – Mauro ALLEGRANZA Aug 29 '17 at 18:43
  • @MANMAID I am not doubting that it is true, but how can we take it as a definition? If you have to explain to me why it is true, isn't it a theorem? – Heuristics Aug 29 '17 at 18:49
  • Definition of P(A) = # of possible occurances of A/# possible occurances. So by definition P(B|A) = # of possible occurances of A and B/# possible occurances of A. Okay. You are right, it does need to be a theorem and not definition that # of possible occurences of A and B/# occurences of A = (#possible occurences of A and B/#total occurences)/(#possible occurences of A/#total occurences). But that is so very trivial it really can not be considered a logical infraction. – fleablood Aug 29 '17 at 18:51
  • On second thought, how exactly is P(A) defined? The intuitive # of A/# of everything is niave and not "real world". So if we don't have a "real world" interpretation of P(A) we also do not have a real world interpretation of P(B) given A. It wouldn't be inconsistent to define both by definition. – fleablood Aug 29 '17 at 18:56
  • Related: https://math.stackexchange.com/questions/2409954/how-to-explain-this-kind-of-reasoning/2409962#2409962 – Ethan Bolker Sep 07 '17 at 14:29
  • This might be relevant. – Mikhail Katz Sep 13 '17 at 15:55

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A possible answer to both of your questions is "trial and error". To understand this viewpoint it is helpful to think of mathematics as one of the exact sciences rather than being a field of investigation apart from the sciences. In the exact sciences one builds models and then tests them against the "real world" to see if they predict meaningful results. Similarly in mathematics we have no apriori reason to expect that this or that type of calculation will necessarily have this or that meaning in the "real world"; rather, trial and error have shown that certain mathematical theories are indeed applicable in the sense of producing seemingly meaningful results.

To answer your second question: in the framework just outlined, mathematical models indeed do not automatically translate to the real world. Through trial and error, some of them have been found to be useful. Others have not (take for example the mathematical entity called a nonmeasurable set and the theorem asserting that a unit ball can be broken up into 5 pieces which, when rearranged, produce two unit balls).

In fact, Bayesian statistics is an excellent case in point: half the experts think it is applicable to describing things in the "real world" and the other half think it is useless. This indicates that the existence itself of a coherent mathematical theory is no guarantee of its "translatability to the real world".

Mikhail Katz
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  • +1 But I think your last paragraph on Bayesian statistics is too short to be useful and probably not expandable here. The real problem (in my opinion) defining of "probability" in the "real world" for one time events is very subtle. You can't just count favorable cases as a fraction of total cases. – Ethan Bolker Sep 07 '17 at 14:33
  • @Ethan, what you seem to be saying is that the debate opposing Bayesians and non-Bayesians involves subtle issues. In fact I personally lack the expertise in this area to offer a meaningful opinion but the fact that the debate involves subtle issues is precisely what I am pointing out. Namely, the applicability of a mathematical theory is not something that can be taken for granted. – Mikhail Katz Sep 07 '17 at 14:36
  • I lack that expertise too, but I follow Andrew Gelman's blog on Statistical Modeling, Causal Inference, and Social Science. It often very informative and often funny. Here's one link from a search there for "bayesian": http://andrewgelman.com/2017/04/05/not-hard-move-away-hypothesis-testing-toward-bayesian-approach-embracing-variation-accepting-uncertainty-actually-approach-easier-approach/ – Ethan Bolker Sep 07 '17 at 14:43
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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?