What is the intuition behind $P(A\cap B) = P(A)P(B)$, given A and B are independent events?
I saw the derivation but wasn't able to to think of it intuitively. Is there any combinatorial reasoning behind it?
I have even seen it for cases like tossing a coin and it works! But I am not convinced that it should be true always. I am looking for a strong foundation of it.
Good question. It also bothered me a lot and I usually fail to answer this question when teaching independence.
So how can I contribute?
One hint is a pointer to Intuition behind independence & conditional probability where this has already been discussed.
A second hint is my own approach to the issue.
There always will be some definitions one understands (and some one does not understand so easily). In the particular case, the one definition I understand intuitively is the definition $P(A|B) = P(A)$. So I take this as a starting point for my thoughts. The other version I then take as a proposition which I can derive. I accept the fact that there are some propositions which are not intuitive to me (but, hey, I can show them from the definitions).
By turning the line of reasoning upside down my desire for an intuition for the definition you give...vanishes.
It's not actually an answer but it could give you peace of mind. :-)