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I tried to understand the most fundamental foundation of the mathematical definition of probability in the most natural/human way.

(At first, I thought I may have found a proper understanding like this:)

First, we need to abstract the events as set. And we assign some real number to the set by measuring these sets. We assign number because it is human nature to quantify things. Let's denote the measure as m. Then it is also instinct/natural for human to use the ratio of m(part for E) / m(total) to measure the probability of event E.

In short, probability is nothing but the ratio of the measurement between part and total. With this sense, probability is only meaningful in a relative context. We can use arbitrary m as it fits. And the P(S) is always 1 since m(S)/m(S) is always 1. And also, it's easy to understand why we use division to define the conditional probability as below. P(A|B)=P(AB)/P(B), because it is actually this:

P(A|B)=m(AB)/m(B) = (m(AB)/m(S)) / (m(B)/m(S)) = P(AB)/P(B) 

I really want to know if there's any flaw of this understanding.

(But after having discussions here, I came to the following ADDs which is specific to the Mathematical Theory of Probability.)

ADD 1

Is there any authoritative definition of what probability is? I found almost all books define the probability based on the 3 famous axioms. But those axioms don't define what probability is. They merely say how probability should behave.

ADD 2

On a second thought, I think I need to add some clarification. We must differentiate between mathematical probability and the interpretation of natural probability. What I mentioned above is my attempt to explain the rational behind the mathematical probability. The natural probability is just a vague concept without precise quantification. In order to make it mathematically operable, we have to do some construction. And the above is what we have done.

ADD 3

As I read the book "Probability and Statistics". It says:

...Almost all work in the mathematical theory of probability...has been related to the following two problems: (i) methods for determining the probabilities of certain events from the specified probabilities of each possible outcome of an experiment and (ii) methods for revising the probabilities of events when additional relevant information is obtained.

So, it occurs to me that "mathematical theory of probability" cannot provide us with the initial probabilities of all outcomes, these initial probabilities have to be specified in some other ways which may come from different interpretations of probability or practical choice or even subjective initiatives. They are represented as various p.d.f/p.f, some of which are quite obscure.

What "mathematical theory of probability" can provide is just methods to calculate the probability of events of interest based on the foundation of those initial probabilities.

So it is once again proved the ideology of mathematics that it doesn't care about what a mathematical object is. But cares about how to manipulate it.

But, despite that the concept of probability is highly controversial and there're so many in-compatible operational interpretations for it, it is very interesting why all authorities agree on a single mathematical theory of probability as the method mathematical manipulation. Are they out of options?

Here's another question about the justification of mathematical theory of probability.

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closed as unclear what you're asking by rschwieb, anorton, Lost1, M Turgeon, Paul Jan 29 at 2:47

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"Then it is also instinct/natural for human to use the ratio of m(part for E) / m(total) to measure the probability of event E." This does not seem instinctive or natural to me. –  Shahab Jan 29 at 1:12
    
By using the same measure m, we are able to compare the E and S on the same basis. By using the division/ratio, we are able to see some kind of relative relations between E and S on that basis m. If we consider S as a certain thing to happen. Then we should get some impression from the ratio regarding how likely E will happen on that basis m. –  smwikipedia Jan 29 at 1:16
    
You probably want to read something about Dempster-Schafer theory en.wikipedia.org/wiki/Dempster%E2%80%93Shafer_theory –  Sergio Parreiras Jan 29 at 1:18
    
Seems to me that one would have got a correct impression if all the events are on the same footing. But in nature all events are not equally likely. The likelihood of a young person growing in height is more than an old person growing in height. (Under the assumption that how one has quantified sets (measure m) is fixed) –  Shahab Jan 29 at 1:22
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With reference to your updated question any function which satisfies those axioms is a probability function. So there are infinitely many definitions of probability. Practically depending on the situation one is picked by mathematicians. –  Shahab Jan 29 at 1:26

2 Answers 2

Mathematicians qua mathematicians cannot answer this.

Kolmogorov defined probability using measure theory: You have a space whose total measure is $1$, and the probability of a subset is the measure of that subset. The rest of mathematical probability theory follows from that as a logical consequence.

However, there are concepts of probability that are not a part of mathematics, and Kolmogorov's theory is a proposed way to model them mathematically. Whether it's the right way is not a mathematical problem.

One commonplace notion of probability would say that "There's a 30% chance that you get this disease if you like this TV show" means that 30% of those who like this TV show get this disease. That's the frequentist interpretation. By that understanding of probability, one cannot say that there is a 50% chance that there was life on Mars a billion years ago, since it makes no sense to say that that happened in 50% of all cases. A degree-of-belief interpretation of probability, however, would allow such a statement. One can apply Kolmogorov's to either of these two interpretations of probability.

Richard T. Cox's book Algebra of Probable Inference is an attempt to justify the application of conventional rules of mathematical probability in the degree-of-belief context.

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2  
I can't parse the phrase "Kolmogorov's theory is a proposed was to model them". –  Byron Schmuland Jan 29 at 2:01
    
@bof Ah! Thanks for that. I am a bit thick-headed today. –  Byron Schmuland Jan 29 at 2:17
    
As I read from the book "Probability and Statistics" by Morris H. DeGroot. There're 3 different interpretations of probability. And they are not compatible. But no matter which one is adopted, mathematical probability theory is the common tool to use. And such a tool seems to be the ultimate thing mathematics can offer. –  smwikipedia Jan 29 at 3:33
    
I updated my question. –  smwikipedia Jan 29 at 4:39
    
@smwikipedia : Did DeGroot say there are ONLY three? There are frequentist views and Bayesian views, and among the latter one has various sorts of subjective probability and also logical epistemic probability. Are those the three kinds DeGroot named? –  Michael Hardy Jan 29 at 20:25

We assign number because it is human nature to quantify things.

Animals often behave as if they understood probability theory (of course, what's really going on is that they've to estimate certain probabilities and behave appropriately). So, I don't think it makes sense to explain probability theory in terms of human biases. Perhaps it can be explained as a byproduct of evolution, but even this is going to be hard, since good evolutionary models tend to be grounded in probabilistic ideas themselves.

Also, I don't the question of what probability is makes a lot of sense. In the modern-day approach to mathematics, we don't typically define what a number "is" or what probability "is," rather we define what a natural number system $(\mathbb{N},0,1,+,\times)$ is and/or what a probability space $(\Omega,\mathcal{E},\mathbf{P})$ is, and then give some motivating examples, like $\mathbb{N} \mapsto \omega$, $0\mapsto\emptyset$, $1\mapsto \{\emptyset\},$ etc.

So a better question would be: why do we care about probability spaces specifically, as opposed to other kinds of mathematical structures that could be used to reason about uncertainty, chance and odds.

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