When is this ever the situation?...Can you give an example of "driving the probability away from 50%"? I note that no one responded to my earlier request for such an example.

No one can give an example because it is logically impossible for it to be the situation, it's not just rare. It cannot be that "All we know is that our hypothesis is either true or false." because to know that something is a hypothesis entails knowing more than nothing. It's like saying "knowing that a statement is either false or a paradox, but having no information at all as to whether it is false or a paradox".