How is independence established and applied in probability? Events $A$ and $B$ are said to be independent if and only if $$\mathbb{P}(A\cap B)=\mathbb{P}(A)\cdot \mathbb{P}(B).$$ There are also well-known definitions of a tuple of events being pairwise or mutually independent (which are not equivalent).
In some elementary texts, I have seen that it is claimed without justification that $A$ and $B$ are independent (often in a context where such a claim is sufficiently obvious to not raise questions, like a dice roll and a coin toss which do not "affect" each other), and then the equation in the definition is used to compute $\mathbb{P}(A\cap B)$ by instead computing $\mathbb{P}(A)\cdot \mathbb{P}(B).$ However, it seems to me that before we can claim that $A$ and $B$ are independent, we need to prove that the equation holds, which just might involve computing all three quantities, thus rendering the concept of independence as baggage in this scenario.
Questions:


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*Are there implicit ways of proving independence that are applicable in some general scenarios? By implicit, I mean either establishing $\mathbb{P}(A\cap B)=\mathbb{P}(A)\cdot \mathbb{P}(B)$ without computing the three probabilities, or establishing independence without proving this equation. If so, what are they? In these two cases, it would then make sense to me that the equation could be used as a consequence of having proven independence, making the concept of independence useful in that scenario.

*If the answer is negative (so implicit methods are unknown), then I have to wonder what purpose this abstraction of independence serves in general. Does it maybe facilitate some proofs that are unrelated to independence, such as restricting to independent cases first and then using it to generalize the argument? Or is independence a  concept that exists for its own sake?


Restricting the answer to discrete/finite probability spaces is not a problem.
 A: My personal take is that probability theory is just a framework that tells you how to calculate things under certain assumptions.  E.g. if you are flipping two fair coins, in most settings independence will be assumed.  And given this assumption, the theory tells you how to calculate certain values, e.g. $P(H_1 \cap H_2) = P(H_1) P(H_2) = 1/4$.
Now, you're asking how we know the two flips are independent.  But let me ask you first, how do you know the coins are fair?  It is just as much an assumption (or part of the model) as independence.
Of course, experimentally we can flip the first coin a billion times and see that the fraction of heads is about half, and do that to the second coin too, to give some evidence for each coin's fairness.  Then you can flip both coins together a billion times, and see that the fraction for each possible outcome is about $1/4$, to given some evidence for independence.  My point is simply that if you ask for "proof" of independence, then it makes sense to also ask for "proof" of fairness... and in many settings it makes sense to ask for neither.
BTW this doesn't even get into the philosophical argument of whether randomness "really exists" etc.  Probability theory says, if you model the coins as fair, and independent, then $P(H_1 \cap H_2) = 1/4$.  It doesn't say anything about the physical existence of such things as fair, independent coins.
Another way to look at independence (indeed any math concept) is to see if it's useful.  Is it a useful concept?  Can you prove interesting theorems about it?  The answer is definitely YES in case of independence.

ADDENDUM based on OP's comments below. 
You're looking for a non-trivial theorem of the form "if some precondition blah is satisfied, then $A,B$ are independent" which is then used to evaluate $P(A \cap B) = P(A) P(B)$.  Here is an example, but I'm not sure it meets the bar of being non-trivial.
Theorem: If $X,Y$ are independent r.v.s, and $f, g$ are any functions, then $f(X), g(Y)$ are independent.
You can use this theorem to evaluate $P(f(X) = a \cap g(Y) = b) = P(f(X)=a) P(g(Y) = b)$.  However, the theorem is proved by checking this defining equation.  So if you didn't prove the theorem, and you don't know how to prove it, but you are just a user, does it count as a non-trivial example?
A: I think that you can say that two events are indipendent without proving it if you have no data about it but logically it should be the case , for example with the events : tomorrow will be a sunny day and tomorrow there will be more than $10000 $ birth in USA. If you have some data you can take them to prove the indipendence/dependence of the events. More and more data you have more and more your prediction will be accurate. Besides , you can simplify things : you can take an event of the real world like 'throwing a real die' and make it in abstract terms so that the event will become 'throwing a theoretical die' which has uniform probability distribution and indipendent outcomes from one throwing to the other. This act of simplification can be useful to model the real event.
