It is widely understood that statistical correlation between two variables ≠ causation. But despite this admonition, people are routinely overconfident in claiming correlations to support particular causal interpretations and are surprised by the results of randomized experiments, suggesting that they are biased & systematically underestimating the prevalence of confounds/common-causation. I speculate that in realistic causal networks or DAGs, the number of possible correlations grows faster than the number of possible causal relationships. So confounds really are that common, and since people do not think in DAGs, the imbalance also explains overconfidence.
You're missing a 4th possibility. A & B are not meaningfully linked. This is very important when dealing with large sets of variables. Your measure of correlation will have a certain percentage of false positives, and discounting the possibility of false positives is important. If the probability of false positives is 1/X you should expect one false correlation for every X comparisons.
And we can’t explain away all of this low success rate as the result of illusory correlations being throw up by the standard statistical problems with findings such as small n, sampling error (A & B just happened to sync together due to randomness), selection bias, publication bias, etc. I’ve read about those problems at length, and despite knowing about all that, there still seems to be a problem: correlation too often ≠ causation.
Full article: http://www.gwern.net/Causality