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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
2  
By the way, this is a well-known problem: en.wikipedia.org/wiki/Bertrand's_box_paradox –  Cocopuffs Nov 12 '12 at 8:09

1 Answer 1

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).

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But the question you phrased (same color) is different. We were asked about the probability that the second one is also white. Maybe its not important. Can you please clear my mistake? Thank you –  Yotam Nov 12 '12 at 7:39

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