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Because you know that you picked gold initially. The odds of the second coin being gold is the odds that you didn't pick 1/3 boxes with with both gold and silver coins, meaning 2/3. The only way the second coin isn't gold is that the initial choice was the box with both silver and gold coins in it, the number of silver coins in that box do not matter because of the precondition of having picked a gold coin.
Of course they matter, they increase the chance that the gold picked in round one was from the double gold box dramatically, which itself hugely increases the odds of round 2 is also gold.
They dont matter because the question is conditioned on that we already picked a box with a gold coin.
The question is what the odds are that we picked the box with both gold and silver, given that we have a box with at least a gold coin in it. There is 1/3 with gold and silver, hence the probabilty of the second coin being gold is 2/3. You could increase the amount of silver coins by infinity and it wouldn't matter. You're picking boxes, not coins.
Yes. But the fact that w already picked a box with the gold coin tells us that it was almost certainly the double gold box and therefore that the probability of the gold second coin is even higher.
No, it tells us nothing. The question is conditioned on a gold coin having been picked.
We didn't pick a box at random, the gameshow host did and revealed a gold coin.
I don't think this is the problem statement, which says:
I don't see anything about a game show host doing anything. Obviously, I agree that the problem turns out differently if a game show host is involved and able to make a choice at any of the steps.
You're right, I got lost in my own example and convinced myself of my wrong intuition which was correct for the original amount of coins.
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And if you picked a gold coin you almost certainly picked it from the double gold bucket…I’m unsure what the issue is here.
You almost certainly did if there actually was a genuine round one, but there wasn't. It was defined that the coin drawn was gold.
Imagine it like this: you have three boxes but the game starts with the gameshow host walking up to a box, looking into it and picking out a gold coin. Does it matter how many silver coins there are?
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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