And the answer, of course, cannot be both .5 and .99. Something has to give.
That is, unless there is an ambiguity hidden in the question, where that ambiguity can be resolved in one of two ways yielding answers of 0.5 and 0.99. And there is. The word “probability” can be expanded out into a definition in terms of set measure. By bringing in cloning, you’ve introduced an ambiguity: it can either be measure over worlds, or measure over (world,observer) pairs.
When you add an observation (“you see X”), you are either excluding world-observer pairs in which the observer has not seen X, or else excluding worlds in which no observer ever sees X. When you say that the probability that it’s tails is 0.5, that’s because it’s tails in half of the worlds. When you say that the probability is 0.99, that’s because it’s tails, that’s because it’s tails for 0.99 of the world-observer pairs.
That is, unless there is an ambiguity hidden in the question, where that ambiguity can be resolved in one of two ways yielding answers of 0.5 and 0.99. And there is. The word “probability” can be expanded out into a definition in terms of set measure. By bringing in cloning, you’ve introduced an ambiguity: it can either be measure over worlds, or measure over (world,observer) pairs.
When you add an observation (“you see X”), you are either excluding world-observer pairs in which the observer has not seen X, or else excluding worlds in which no observer ever sees X. When you say that the probability that it’s tails is 0.5, that’s because it’s tails in half of the worlds. When you say that the probability is 0.99, that’s because it’s tails, that’s because it’s tails for 0.99 of the world-observer pairs.