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So what would you say is your probability of withdrawing a gold coin if everything else is the same, but the third box has one gold coin and 10 silver coins, instead of just one gold and one silver coin?
Of withdrawing another gold coin? 2/3
Why 2/3?
OP's intuition says that once you pick one gold coin, you know that you have one of two boxes, and that there are exactly 12 coins in those two boxes combined, two of which are gold, so that would put the probability of getting another gold coin at 2/12.
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.
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.
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