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PaperclipPerfector  2mo ago (text post)   2235 thread views

Friday Fun Thread for September 6, 2024

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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.

Wait, I am wrong.

The probability of picking the two-gold box is 1/3. The probability of picking the mixed box and finding gold on the first draw is 1/3*1/11=1/33.

11/33 vs 1/33, I am 11 times more likely to find another gold coin, not two times.

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.

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:

You reach your hand into one of the boxes

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.

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.

More comments

I think this checks out but someone else will need to check for me.

It's probably good that you're not trying to flex too much about how smart you are due to finding the solution to this problem incredibly obvious, because it seems that you got the answer correct the same way that a broken clock gets the time correct twice every day. ;)

Okay, rude.

I disagree. Maybe this is the reason I "always forget" the simple route; because I'm not sure it's actually right. I did this two different ways, my renormalization route (thinking of things as a tree with info sets) and just brute reproducing the wiki entry on using Bayes to solve it.

Method 1: Renormalization

There's a 1/3 chance of picking each box, one which has a 100% chance of giving you a gold on the first draw and the other has a 1/11 chance (ignoring the option with zero chance of getting a first gold), so the chances of me being in each relevant box at the current state are 1/3 and 1/33. To renormalize, I need to multiply by the reciprocal of their sum, 1/3 + 1/33 = 12/33.

So my chance of being in the GG box is 11/12 and my chance of being in the G10S box is 1/12.

Method 2: Straight Bayes, yo

Just shutting up and calculating, reproducing the wiki article directly.

P(GG|see gold) = P(see gold|GG)*(1/3) / [P(see gold|GG)*(1/3) + 0 + P(see gold|G10S)*(1/3)]

P(GG|see gold) = (1/3) / (1/3 + 0 + 1/33)

= (1/3) / (12/33)

= 11/12

I'd say law of conditional probability is the simplest route here. P(2nd Coin Gold | 1st Coin Gold) = P(Both Gold) / P(1st Coin Gold) = 1/3 / (1/3 * 1 + 1/3 * 1/11) = 1/3 / (1/3 + 1/33) = 1/3 / (12/33) = 1/3 * 33/12 = 11/12.

This is obviously correct, I have no idea what the other people are saying.

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