I'm having a discussion with a friend about "I think not(X)" vs. "I don't think X". Is the former correct or not? For example, are these correct?

"I think I didn't mention this yet."
"I think I don't stand a chance."
"I think I can't do this."
"I think I won't go."

These sound correct to me, but maybe I'm wrong? Do you know of a rule which says that in a sentence with "think", you should only negate "think", and not the statement that follows (ie. "I don't think I mention this yet", "I don't think I stand a chance", "I don't think I can do this.", "I don't think I'll go.").
They are correct, but not the normal formation. We tend to negate the main clause (this is called transferred negation) after verbs such as think and suppose.
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Thanks, I didn't know this was called "transferred negation", that's a good pointer in Google :) According to Section 6 in this doc: http://www.lancs.ac.uk/postgrad/xiaoz/papers/negation%20in%20english.doc , the "I don't think X" formulation has a slighly different use, as the strength of the negation is reduced. So "I don't think X" is more suitable in polite conversation, while "I think not(X)" is more suitable when making an argument. What do you think?

"(35) a. I think I haven’t got a lot there no, that’s not a lot. (BNC: KCP)
b. I wasn’t I? and I don’t think I have any, I don’t know. (BNC: KD0)

Transferred negation is essentially a type of indirect negation, which indicates a reduced ‘strength scale’ of negation (Horn 1978a). As Shen (1989: 4) observes, the degree of negation becomes lower when the negator is further away from what is negated. As such, (35a) expresses a stronger attitude than (35b). The indirectness and reduced degree of negation also account for why transferred negation typically co-occurs with verbs of opinion and perception, which is appropriate as a politeness strategy but ineffective in argumentation."
That sounds like a good guideline; it is quite logical. This source seems to be a student research paper, however, and research papers tend to belabour their points. I wouldn't rely solely on this guideline for the interpretation of opinions.