Does independence necessarily mean uncorrelatedness? We all know that two independent events are uncorrelated, don't we?
Nonetheless, we can find events that are correlated, yet they are independent such as the examples found on the Suprious Correlations website$^*$.
Is there a problem here or is it just me who's missing a ring in the chain?
$^*$ The website is actually meant to show that correlation doesn't imply causality but I think we can agree that the variables shown are independent (i.e., the occurrence of one does not affect the occurrence of the other).

You should keep stochastic independence distinct from causal independence. 
Two random variables that are stochastically independent are uncorrelated by definition. 
Two random variables that are causally independent ($A$ does not imply/causes $B$, nor vice versa) may be correlated. 
It is also possible that some third random variables $C$ separately influences both $A$ and $B$, making them correlated. 
