I think I have no way of assigning numbers to the quantities P(causation|correlation) and P(causation|~correlation) assessed over all examples of pairs of variables. If you do, tell me what numbers you get.

My original question was whether you think the probabilities are equal. This reply does not appear to address that question. Even if you have no way of assigning numbers, that does not imply that the three possibilities (>, =, <) are equally likely. Let's say we somehow did find those probabilities. Would you be willing to say, right now, that they would turn out to be equal (with high probability)?

I asked why and you have said "intuition", which means that you don't know why.

Okay, here's my reasoning (which I thought was intuitively obvious, hence the talk of "intuition", but illusion of transparency, I guess):

The presence of a correlation between two variables means (among other things) that those two variables are statistically dependent. There are many ways for variables to be dependent, one of which is causation. When you observe that a correlation is present, you are effectively eliminating the possibility that the variables are independent. With this possibility gone, the remaining possibilities must increase in probability mass, i.e. become more likely, if we still want the total to sum to 1. This includes the possibility of causation. Thus, the probability of some causal link existing is higher after we observe a correlation than before: P(causation|correlation) > P(causation|~correlation).

There is no such thing as a correlation not "present in the data".

If you are using a flawed or unsuitable analysis method, it is very possible for you to (seemingly) get a correlation when in fact no such correlation exists. An example of such a flawed method may be found here, where a correlation is found between ratios of quantities despite those quantities being statistically independent, thus giving the false impression that a correlation is present when it is actually not.

What observations would you undertake to determine whether a correlation is, in your terms, a "real" correlation?

As I suggested in my reply to Lumifer, redundancy helps.