TITLE: What is the meaning of "independent events " and how can we logically conclude independence of two events in probability?
QUESTION [6 upvotes]: What is the meaning of  "independent events " in probability

For eg: Two events (say A and B)are independent , what I understand is the occurrence of A is not affected by occurrence of B .But I am not comfortable with this understanding , thinking this way every events I meet upon are independent!, does there exist a more mathematical definition( I don't want the formula associated with it)

Another thing I want to know in a real case or a problem how do we
  understand(logically) if two events are independent?


That is without verifying that $P(A\cap B) = P(A)P(B) \Large\color{blue}{\star}$  how do we conclude that two events $A$ and $B$ are independent or dependent?.
Take for eg an example by "Tim" 

Suppose you throw a die. The probability you throw a six(Event $A$) is $\frac16$ and the
  probability you throw an even number(Event $B$) is $\frac12$.

And event $C$ such that $A$ and $B$ both happen would mean $A$ should happen(As here $A$ is a subset of $B$) hence it's too $\frac16$
My thought:
The above example suggest something like this "Suppose I define two events $A$ and $B$ and if one of the even is a subset of the other then the events are not independent". Of course such an argument is not a sufficient condition for dependency , for eg: consider an event $A$ throwing a dice and getting an odd number and event $B$ getting 6. They aren't independent ($\Large\color{blue}{\star}$ isn't satisfied) . So again I improve my suggestion "Suppose I define two events $A$ and $B$ and if one of them is a subset of the other or their intersection is a null-set then the events are not independent $\Large\color{red}{\star}$"
So at last is $\Large\color{red}{\star}$ a sufficient condition?. Or does there exist a suffcient condition (other than satisfying the formulas $\Large\color{blue}{\star}$? And what is the proof, I can't prove my statement as the idea for me is not that much mathematical .

REPLY [3 votes]: As pointed out, the question you pose is both mathematical (in which case the answer is clear and unambiguous) and philosophical (in which case, it isn't).
For a starter on the philosophy of probability - start here.
Your statement $\Large\color{red}{\star}$ is a sufficient condition is correct but it is highly restrictive. The statement $\Large\color{blue}{\star}$ is a necessarily and sufficient condition - indeed the necessarily and sufficient condition.
Consider the two events "Arsenal will win their next game" and "Manchester United will win their next game". Are these events independent? If they are playing each other - clearly not, if they are playing other teams then clearly yes.
So there exists a third event "Arsenal are playing Manchester United in their next game". The first two events are clearly dependent on this event.
The probability of this third event is interesting because it strikes at the heart of how one interprets probability.
This probability is either 0 or 1 - I could go online now and find out. But if I do not know -  can I determine a probability for it that is not 0 or 1. If I can, how can this be interpreted? Further, what if nobody knows (e.g. the draw has not yet been determined)?
For some interpretations of this state of indeterminacy see Schrodinger's Cat
I think the point is that mathematics is mathematics, physics is physics, music is music and dance is dance - what they mean is down to humanity.