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Notes -
You cannot look into the box, you cannot weigh the boxes, or shake them, or anything like that, so the situation is not equivalent to a gameshow host looking into the box and deliberately rummaging through and picking the one gold coin that is there.
Let's turn up the amount of silver coins. Imagine there are 1000 silver coins and one gold coin in the third box. You pick one coin out of a box and it turns out to be gold. Which is more probable, a) that you happened to pick the one and only gold coin from a 1001 coins, or b) that you picked one gold coin from a box that contains only two gold coins?
Remember, there are a 1000 ways to pick a silver coin from the third box, in which case we don't end up in the scenario where we are picking a second coin. It is only the very rare occurrence when we happen to pick the third box and pick the one gold coin out of a 1001 coins that we are in the scenario where picking a second silver coin is possible.
You're right. I got the right answer by accident and then started convincing myself my explanation was the right one.
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