I think this is not correct. If the oracle is queried whether something happens that it can not know, it will always (have to) give a random answer, the thing with hitting the probability p exactly is misleading. The paper gives the example of a touring machine querying the oracle whether the probability of itself halting is smaller or larger than 50 % and then doing exactly the opposite. So the oracle can not tell, assigns a 50 % probability to both halting and not halting and therefore always randomizes the answer. It is really more about the nature of the query, the probability thing is more of a technicality.
As I understand it they are essentially talking about running the oracle for some time and then asking for the current probability of the result being one or zero. If the thing asked is computable at all and in less than the time the oracle ran, then the answer will be certain, otherwise the oracle may have accumulated some evidence and assign different probabilities to both results or may still have no idea after the given time and will still be indifferent.
The question is then, does the oracle believe that the result is X with probability larger than p. If the oracle does not know, it will think the result will be X with 50 %. If you set p above 50 %, you will always get zero, if you set it below 50 %, you will always get one and if you set p to 50 %, then the oracle will randomize. No matter what p is, you won't get any information out of the oracle in that case and that is of course by design because otherwise you could let the oracle solve the halting problem and that would imply such a oracle does not exist.
If you asked the oracle about a Turing machine simulating a die - they are nondeterministic Turing machines with access to randomness - you would have to set p to 1/6 and ask whether the oracle believes the next roll will yield for example a six. But if the outcome is really generated from true randomness, the oracle can not know what the next roll will yield, has itself to assign a probability of 1/6 to rolling a six and will again randomize every time.
As I understand it they are essentially talking about running the oracle for some time and then asking for the current probability of the result being one or zero. If the thing asked is computable at all and in less than the time the oracle ran, then the answer will be certain, otherwise the oracle may have accumulated some evidence and assign different probabilities to both results or may still have no idea after the given time and will still be indifferent.
The question is then, does the oracle believe that the result is X with probability larger than p. If the oracle does not know, it will think the result will be X with 50 %. If you set p above 50 %, you will always get zero, if you set it below 50 %, you will always get one and if you set p to 50 %, then the oracle will randomize. No matter what p is, you won't get any information out of the oracle in that case and that is of course by design because otherwise you could let the oracle solve the halting problem and that would imply such a oracle does not exist.
If you asked the oracle about a Turing machine simulating a die - they are nondeterministic Turing machines with access to randomness - you would have to set p to 1/6 and ask whether the oracle believes the next roll will yield for example a six. But if the outcome is really generated from true randomness, the oracle can not know what the next roll will yield, has itself to assign a probability of 1/6 to rolling a six and will again randomize every time.