I came across these sentences in a learner's dictionary (see examples for sense 3a) today:
Do you think he can still be alive? [= do you think it is possible that he is still alive.]
I don't think he can still be alive. [= I think he must be dead.]
Since the second one is negative, it's probably not wrong, as it's pretty much a paraphrase of I think he can't still be alive. and can't can be used with the sense of 0% probability, i.e. can't = I'm certain it's not true.
From what I've known and what I've read, could, but not can, is usually used this way to indicate probability either in the present or the future, as in he coud be in the garden.
As for possibility, can can only be used to tell general possibilities/truths (It can be rainy here during September.) or the possibility in a particular situation (We can go to Paris this weekend, since we don't have to work).
What I Think & What I've Tried Doing
I think the first one should read
Do you think he could still be alive?
and probably even the second one should read
I don't think he could still be alive.
However, it seemed strange to me (who is a non-native) when I replaced could with may in the first sentence, which is usually possible when talking about probability.
*Do you think he may still be alive?
I also noticed that
Do you think he is still alive?
seemed more natural.
- What is can doing there?
- In what context is can used like this?
- Can could do the job too?
I'm talking about using can and could in the present here. (This may lead to varying answers if I don't make this clear, I think)
Please, please avoid the word "possible" where possible. If you're going to use it, please make it clear what kind of "possible" you mean: generally possible (he can be hard at times), possible in the situation (we can go to Paris this weekend because we don't have to work), possible as in probability (he could be in the car.)