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.)
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.
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.
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.