# Two answers for a conditional probability problem

Here's a question I'm thinking about:

There are three drawers. Drawer A contains 2 black socks, Drawer 2 contains two white socks and the third drawer is mixed. If I pulled a sock from one of the drawers. Given that it is white, what is the probability that the other sock in the drawer is white?

So what I'm thinking is that I picked either the second or the third. If I picked the second I know the other one is white, and if I picked the third I know that the other one is black and so the probability is $1\cdot \frac{1}{2} + 0\cdot \frac{1}{2}=\frac{1}{2}$

I saw someone else's solution and he says that after picking one sock we have a new sample space $$\Omega=\{(W_1,W_2),(W_2,W_1),(W_m,B)$$ where $W_1,W_2$ are the two in the second drawer and $W_m$ is the one in the mixed drawer. Anyway he concluded that the probability is $$P(second\, is\, white)=\frac{\{(W_1,W_2),(W_2,W_1)\}}{|\Omega|}=\frac{2}{3}$$

I think he's mistaken for counting the order in which the socks were pulled, because it is given that we pulled a white one, it doesn't matter which exactly.

So, who is right? And if I'm right, was I also right about his mistake?

Thanks!

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You mean "Drawer A contains $2$ black socks"? – joriki Nov 12 '12 at 7:28
"What is the probability that the other sock in the drawer is white?" is a strange question, since the sock's colour is deterministic. – Cocopuffs Nov 12 '12 at 7:34
@Cocopuffs: That's only under some interpretations of probability. I would argue that an interpretation of probability that doesn't allow us to speak of a probability in this case offers an unduly restricted concept of probabilities. – joriki Nov 12 '12 at 7:38
@joriki: yes that's what I meant – Yotam Nov 12 '12 at 7:40
By the way, this is a well-known problem: en.wikipedia.org/wiki/Bertrand's_box_paradox – Cocopuffs Nov 12 '12 at 8:09

If you were right, then obviously the answer to "What is the probability that the second sock is black, given that the first is black?" would also be $\frac12$.
What would be your answer to "What is the probability that the second sock has the same colour (no matter which) as the first?" - of course $\frac23$ because this holds for two out of three drawers. By symmetry, the answer to this question cannot suddenly change from $\frac23$ to $\frac 12$ if the asker continues his question with "By the way, the first sock is white" (or black or he dies mid-sentence of a heart attack).