I got up in them middle of the night to write this and it was all muddled. Then I got up in the middle of another night thinking 'This makes no sense!' And it didn't, because I had written P(causation|corrrelation) = P(causation) - P(correlation|no causation). Stupid me. I'll correct it later. In short: They're not independent, as correlation is a prerequisite for causation. P(correlation) is not 0 or we wouldn't be discussing P(correlation) with such fascination. I'll explain it with more rigor when I'm not really tired and needing to get up in seven hours.

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I occasionally see misuse of the mantra 'Correlation does not imply causation'. So I don't have to write all this every time I see it, here's a concise explanation:

(Note on notation for non-mathy people: P(x) is the probability that x is true. P(y|x) means the probability that y is true given that x is true.)

The source of the confusion with this statement is that 'imply' can mean either 'entail' or 'suggest'. 'x entails y' means P(y|x)=1). 'x suggests y' means P(y|x)>P(y).

In 'correlation does not imply causation', the word 'imply' is being used to mean entail. Correlation does suggest causation.

Also, if you've ever been tempted to claim that 'lack of evidence is not evidence of lack', I encourage you to apply the same [deleted because I muddled it] reasoning, keeping in mind that the terms 'evidence' and 'proof' are not typically considered interchangeable.

I argue by analogy a lot (though less than I used to). People are stupid, and analogies provide a very pithy way of demonstrating why what they just said is really, really stupid. Alas, they're pithy because the working premise is implicit, and they're doubly ineffective when they mention Hitler. Consider this discussion I've had ever so many times:

Foo: Bar punched me in the face just for kicks. He's mean.
Baz: He's my friend. You're a jerk for calling him mean. He's nice to me.
Me: Hitler was nice to his friends.
Baz: You're saying punching Foo the face is as bad as genocide?

A half-way intelligent person would realize the argument by analogy, made lucid, is this:

Your argument is premised on the claim that 'If X is nice to his friends, he's not a mean person.' However, consider the following argument using that same principle:
1) If X is nice to his friends, he's not a mean person.
2) Hitler was nice to his friends.
3) Therefore Hitler was not a mean person.

Clearly, the conclusion, 3), is false. 2) might be true. I suspect it that is--if you believe otherwise, just consider hypothetically that it were. Supposing 2) is true, do you believe 3) is therefore true? Probably not, which entails that 1) must be false (via denying the antecedent).

Thus, completely independent of whether punching Foo in the face is mean--or in any way comparable to what Hitler did¹--the fact that Bar is nice to you, his friend, does not entail that he isn't mean.

¹ Neither your argument nor mine is about what Hitler did or what Bar did to Foo. Yours is about whether Foo is a mean person and how he behaves toward you, not Foo, and mine is about whether your reasoning is sound.

That's a really, really long-winded, and any half-way intelligent person would be perfectly capable of getting it all from the six word reply I gave in the initial example.

I, however, do not frequently find myself replying to half-way intelligent people, as they would not say something so stupid as Bar did.

Since analogy is about as effective with these people as saying 'You're stupid', I sometimes go with the latter. An intelligent person will recognize that as shorthand for the relevant analogy.

(In the future, however, I may just say 'Hitler!' with a link to this post.)