The simple answer is that your choice is also probabilistic. Let’s say that your disposition is one that would make it very likely you will choose to take only box A. Then this fact about yourself becomes evidence for the proposition that A contains a million dollars. Likewise if your disposition was to take both, it would provide evidence that A was empty.
Now let’s say that you’re pretty damn certain that this Omega guy is who he says he is, and that he was able to predict this disposition of yours; then, noting your decision to take only A stands as strong evidence that the box contains the million dollars. Likewise with the decision to take both.
But what if, you say, I already expected to be the kind of person who would take only box A? That is, that the probability distribution over my expected dispositions was 95% only box A and 5% both boxes? Well then it follows that your prior over the contents of box A will be 95% that is contains the million and 5% that it is empty. And as a result, the likely case of you actually choosing to take only box A need only have a small effect on your expectation of the contents of the box (~.05 change to reach ~1), but in the case that you introspect and find that really, you’re the kind of person who would take both, then your expectation that the box has a million dollars will drop by exactly 19(=.95/.05) times as much as it would get raised by the opposite evidence (resulting in ~0 chance that it contains the million). Making the less likely choice will create a much greater change in expectation, while the more common choice will induce a smaller change (since you already expected the result of that choice).
The simple answer is that your choice is also probabilistic. Let’s say that your disposition is one that would make it very likely you will choose to take only box A. Then this fact about yourself becomes evidence for the proposition that A contains a million dollars. Likewise if your disposition was to take both, it would provide evidence that A was empty.
Now let’s say that you’re pretty damn certain that this Omega guy is who he says he is, and that he was able to predict this disposition of yours; then, noting your decision to take only A stands as strong evidence that the box contains the million dollars. Likewise with the decision to take both.
But what if, you say, I already expected to be the kind of person who would take only box A? That is, that the probability distribution over my expected dispositions was 95% only box A and 5% both boxes? Well then it follows that your prior over the contents of box A will be 95% that is contains the million and 5% that it is empty. And as a result, the likely case of you actually choosing to take only box A need only have a small effect on your expectation of the contents of the box (~.05 change to reach ~1), but in the case that you introspect and find that really, you’re the kind of person who would take both, then your expectation that the box has a million dollars will drop by exactly 19(=.95/.05) times as much as it would get raised by the opposite evidence (resulting in ~0 chance that it contains the million). Making the less likely choice will create a much greater change in expectation, while the more common choice will induce a smaller change (since you already expected the result of that choice).
Hope that made sense.