I Taking socks out of drawers, conditional probability

[FactChecker]

first I need to think of my sample space, then I would think what probability model I'd like on the elements of the sample space, in my case in is uniform model,

I think this is your fundamental error. The probability distribution of the sample space is not always uniform. It is up to you to calculate the correct probabilities using a valid mathematical process.
This is a very common situation, where the process presented is done in a sequence of steps (pick a drawer, then pick a sock from the drawer). The conditional probabilities of the results of the individual steps are clear. Use those to calculate the correct probabilities of the sample space. Your assumption that the sample space has a uniform distribution is just wishful thinking.

 

[CGandC]

You are mistaken, it is you that is "trying to jump straight into the deep waters". I have shown you the way this material is taught to more than a million 14 year olds in the UK each year; IME most of them understand it.

I've been taught this material in a school also and I understood it. But in school it is learned in a matter which doesn't make you think about what you're doing, why you're doing it and where do these steps stem from, these are research questions one intends to find answers in a place of scholarship like the academia.

You seem to be missing the important point here but I've tried explaining it to you but you keep ignoring it.
I will give you another example that can relate to the way you're viewing the current discussion: A kid knows that if an object drops down then a force of gravity is acting on it and that its weight is $w = m \cdot g$. One who learns physics in the the academia will look again at the problem of the object falling down but will try to answer it in a much more research and rigorous oriented way, for example, he'll make a proper free body diagram, define coordinates, define the vectors, define the assumptions, maybe look at the energy expression of the object, etc... In short, he'll try to dive deeper into the details in order to gain a better understanding of the subject which will in turn merit him later on when he will deal with more complicated problems/situations concerning the area of his study, that is - the student is intended to expand his ways of thinking about those details that concern his field of erudition.

I hope I convinced you, If so then I hope it will make you a virtue; if you still remain unconvinced and continue to be the discerning critic of everything one has to say then there won't be any agreement between us as to the idea of how one should look at problems ( which, I say, sometimes there are isn't a "single" way to look at them and the many efforts of differentiation and attempts of understanding the convoluted paths to establish a clear understanding should also be appreciated ).

I think this is your fundamental error. The probability distribution of the sample space is not always uniform. It is up to you to calculate the correct probabilities using a valid mathematical process.
This is a very common situation, where the process presented is done in a sequence of steps (pick a drawer, then pick a sock from the drawer). The conditional probabilities of the results of the individual steps are clear. Use those to calculate the correct probabilities of the sample space. Your assumption that the sample space has a uniform distribution is just wishful thinking.

Thanks, I think that's enough for now. I'll try to unthink of this problem for the mean-time and approach to it later; and if that was not helpful then I'll discuss with my professor.

 

[FactChecker]

I often like to use extreme examples for "sanity checks" of my thinking.
Suppose you had 10 drawers. Nine have one black sock each and the tenth has a million white socks. You pick one drawer out of the ten with uniform probability and then pick a sock from that drawer with uniform probability. Even though the vast majority of socks are white, their probabilities add up to only 1/10.

 

[phinds]

I often like to use extreme examples for "sanity checks" of my thinking.

Yep. Often clarifies things quite nicely and usually with very little effort.

 

[vela]

How do I learn from basics if you're letting me jump straight to deep waters by thinking of probabilities? ( not the the problem's hard, I'm talking about establishing an understanding ) first I need to think of my sample space, then I would think what probability model I'd like on the elements of the sample space, in my case in is uniform model, then I'd start thinking about probabilities but only after I've combinatorically reasoned about the problem?

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.

 

[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.