I don't think so, do you?

Happy new year!  Here's a simple but important one.  I hear many Polish students of English answering questions like this:
  
I think yes.
I think no.
I think I won't pass the exam.


All of these are clear enough--but not natural English.  In the case of the first two examples, native speakers are more likely to say:


I think so.
I don't think so.


Note that we say this only in isolation, that is, if we're only going to make a short response to a question, or a fact or opinion that was already stated.  For example:


--Is it getting colder outside?    --I think so.
--I bet he's got a lot of money.    --I don't think so.


We do not use this form when we're introducing an opinion or a fact.  We don't say:


I think so that it's getting colder outside.


In that case, we'd just take out the word so and say:


I think that it's getting colder outside.  


When we are expressing a negative opinion, that is, saying that we disagree, we are more likely to use this form:


I don't think I'm going to pass the exam.


instead of:


I think I'm not going to pass the exam.


In other words, we move the negative (not) so that it applies to the verb think.