Can one infer independence by simple reasoning/intuition? From my recent experience in probability, it feels as though independence is something we "discover" from the system via the equation: 
$$P(A)*P(B)=P(A\cap B)$$
Could one ever conclude independence from the "system" by intuition? Is it wise to conclude independence for events that are "seemingly" independent? What would be some interesting examples where this would fail. 

An interesting example is as follows
Suppose there are two fair six faced dice, say I and II. Both dice are rolled together, let $X$ denotes the face shown by die I and $Y$ denotes the face shown by die II. 
Define $Z:= X+Y$ and $U := $ remainder when $Z$ is divided by $6 $.
Surprisingly in the above construction random variables $U$ & $X$ are independent. 
For example, let $A= \{X= 1\}  $  & $B=\{ U= 1\} = 
\{ (X,Y)= (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\} $
Then $\mathbb{P}(A)= \dfrac{1}{6}$,  $\mathbb{P}(B)= \dfrac{1}{6}$ and $\mathbb{P}(A\cap B)= \mathbb{P}(\{ (X,Y)= (1,6)\})=  \dfrac{1}{36}= \mathbb{P}(A) * \mathbb{P}(B)$.
