Independent random variables versus non correlated variables

From: Will (william108_at_gmail.com)
Date: 11/16/04 

Date: 16 Nov 2004 02:46:27 -0800

I have a question which I guess revolves around the meaning of
independence of random variables. The definition I am aware of is in
the context of discrete rvs X and Y where for all possible values of x
and y, p(x,y)=p(x)p(y).
So this means that in terms of independence of events, for each
possible combination of events generated by X and Y those events are
independent.

Now, in Statistics we talk about variables being independent, which
means among other things that they are not correlated. I can't see how
this relates to the above "for each possible combination of events
generated by X and Y those events are independent".
Can anyone give me some insight into this?
Thanks so much.

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