Taking socks out of drawers, conditional probability

[CGandC]

I don't think you can achieve what you hope to with this problem. As others have pointed out, for the sample space you constructed, the probabilities are not uniform, so you can't simply count to calculate the probabilities. You have to calculate them using different reasoning.

You might want to consider a really simple case. You toss two unfair coins. The sample space is {HH, HT, TH, TT}. How are you going to reason using combinatorics to derive the probability for each outcome? You can't. You need some other method and use more information to assign probabilities.

While I can understand your desire to solve the problem from first principles, I don't think it's particularly useful here. It would be similar to trying to find the derivative of $\sin x^2$ using the limit definition of the derivative rather than making use of the chain rule.

Yes, I guess you're right. It's one of these problems ( rare ones I have to say because I've done many combinatorial problems by approaching them from a set-theory perspective [ meaning, I've built the sets to be counted themselves and came up with ways to count their elements ], but this one stumbled me using that approach ) that indeed need a different reasoning and not viewing them in a bottom-up fashion, in that case I can immediately find the probabilities as @pbuk and others have suggested and solve the problem.
Thanks very much for the help guys!

 

[FactChecker]

Yes, I guess you're right. It's one of these problems ( rare ones I have to say because I've done many combinatorial problems by approaching them from a set-theory perspective

Then this is not as rare as you have been led to believe so far by simple class problems. There are many, many things that are done in a series of steps where the probabilities at each step are conditional on what happened in the prior steps. (You can make up thousands if you try.) The probabilities of those sample sets are not uniform and can not be determined just by counting the number of elements.