BACKSTORY AND RELEVANT LINKS
So over on Puzzling.SE, a user asked this question. DISCLAIMER: I'm not asking you guys for the answer to the question, but rather an answer to the discussion that ensued about evaluating the logic of a certain answer to the question. I believe that a key part of it isn't correct; the answer-er (obviously) believes that it is, so we started a discussion in the chat.
The problem is that we are going in circles on our discussion - we agree on a lot, but we have these two little areas that we can't see straight on. Could you guys shed some light on which one of us is correct, and why?
NOTE: Please do not address the full question and its answer. If possible, address only the "key part" that I mentioned earlier.
In the original question discussed above, we are told that on a game show, there are 8 boxes, each containing 2 stones. Some (at least half) are precious, and the rest are not precious. The participant selects a box, from which the host selects a stone at random. The stone turns out to be precious. At this point the host (a perfect logician for our purposes) announces that there is a 50% chance that the other stone in the box is precious. The host does know how many boxes there are with two precious stones, one precious stone, and zero precious stones. He does not know which boxes they are.
Now, given that information, both the other answer-er and I conclude that the two possible distributions of precious stones among the boxes is 00111222 and 11112222. I have since realized that this is false. However, with those distributions, our discussion still stands:
We begin to differ on the question "If we swapped the boxes, is it true that we would always have a higher/lower/same(choose one or none) chance of picking up precious pebble next, given the possible distributions". I argue that the probability for this is always higher; he argues that it isn't always higher.
Are either of us correct?