Marble problem 2

When we left them last week, Bengt and Sture were building pyramids out of marbles. Since time in problem-space is independent of time in the real world (and also, I wrote this problem a while ago) they are still working on that, but they are nearly finished. Oh boy, they have built lots of pyramids.

There are twelve marbles left in the bag. Each marble is either black or white. The marbles are otherwise identical. Sture takes four marbles, and then gives the bag to Bengt. Bengt takes three marbles and places them on the ground. They are all white. He reaches in to take one more.

"I wonder what the probability is of the last one being white." says Bengt.
"Well, if those three were white, then there are most likely fewer white left."
"But on the other hand, the fact that these three were white suggests that there were more white ones than black ones when you gave me the bag."
"Hm. Sounds like a good problem."
"I don't know. I think it might be a bad problem, actually."

1. Is it a good problem? Or, more specifically, is it a problem that can be solved? Is it possible to say anything about the probability with the given information, and in that case, what?
   
2. Assume that half of the marbles are white when Bengt gets the bag. Same question.
   
3. Assume instead that half of the marbles are white to begin with, and that Sture takes four at random.
   
4. Assume instead that half of them are white to begin with, but Sture takes four of the same colour.
   
5. Assume instead that half of them are white to begin with, but Sture starts by taking two of the same colour, and then picks the other two at random.
   
6. Assume instead that we don't know the original distribution, but that each marble is originally equally likely to be black or white, and that Sture takes four at random.
   
7. The same, but Sture takes two of each colour.
   
8. The same, but Sture takes four of the most prevalent colour. (If they are equally common he picks a colour at random.)