I feel (linear) correlation is more a statistics concept, less a probability concept. In contrast, Independence has 2 interpretations — in prob vs stats — see other posts in this blog.
In a theoretical model, the color vs the points on a random poker card are independent, so out of 9999 trials, the data collected should show very very low correlation, but perhaps non-zero correlation!
From this example, I feel in a theoretical model, correlation isn’t important. However, in real world statistics, correlation is probably more important than Ind. As described in other blogposts, I feel ind is shades of grey, to be measured … using correlation as the measurement.
Whenever someone says 2 thingies are independent, i think of a logical, theoretical models (probabilistic). In the real world, we are never really sure how independent.
Whenever someone talks about correlation/covariance, i think of statistics on observed data.
It’s well known that 2 (linearly) uncorrelated variables may be dependent !
Many people prefer to day “A and B are uncorrelated” without saying “linear”, when they really mean “they don’t depend on or influence each other”. I feel most of the time the meaning is imprecise and unclear.