A puzzle about reflection

A puzzle.

After a sinful life, a formal epistemologist appears before the gates of heaven. St. Peter takes pity on the fellow, and offers the following deal. God will choose a number in one of two ways. First, God might choose the number randomly from the set of all possible numbers, with each number being equally likely to be chosen. Second, God might choose the number by flipping a coin some number of times -- the number chosen will be equal to the number of heads results before the first tails results. If the epistemologist can guess which method God used the epistemologist can go to heaven. In order to help him make his choice, St. Peter relates to the epistemologist that God's method lead him to the number 2.

Which should the epistemologist choose?

At first glance, it seems like God obviously chose the second method. The probability that God would have chosen 2 using that method is .25.  The probability that God would have chosen 2 using the first method is practically 0. Given the fact that it is vastly more likely that would have chosen 2 with that method, the epistemologist should become much more confident that it is.

What is weird about this answer is that it doesn't really depend upon the particular number chosen. The logic holds just as well if God chose 16, or 10 million, or a googolplex. No matter what, it is vastly more likely that any given finite number will be chosen using the second method than the first. In fact, the second method will produce each number with some positive finite probability, while the first will only produce any number with an infinitesimal probability.

Before St. Peter relates the number God chose, the epistemologist shouldn't feel strongly either way. Afterward, he should become very confident that God chose the second method, no matter what number St. Peter says. The epistemologist should know this, and so the situation should violate van Fraassen's reflection principle: while the epistemologist knows that his future evidence will make a certain judgment rational, that judgment is not rational for the epistemologist at the time.

While reflection principle is controversial, this seems like a very weird violation of it. But one that, as  far as I can tell, is inescapable. Short of denying the possibility of the scenario, it seems impossible to avoid.


(I found this puzzle in a paper by a statistician many years ago, and I have not seen it again in print.)