In the marble example:
To keep the procedure the same as the donkey problem, the host would have to remove 98 black marbles, so you know the white marble is either in your hand or in the bag. You then get the option of re-chosing. At this point there are only two possibilities. Either the marble is in your hand, or in the bag. Since you can re-choose, I still say it is 50/50 at this point. The previous 98 marbles have been removed from the equation. At the beginning we knew that one in the 100 was white, now we know that one in two is white, and we get to choose one.