A parent has 2 children. Given that one is a boy, what is the probability that the other is a girl? (Look for specifics below) So, I've been browsing through the net about this (including this site) and I've stumbling upon different answer. In this site, most of the answer were 2/3 but the arguments here are that BG and GB are different since the order of birth are of consequence. 

However, what if one would not consider the order of birth but only the existence of the said children? What is the probability that there is one boy and one girl in a family with a children of 2 given that one is a boy and having a boy as a child is as likely as having a girl as a child?

Same case as another post here, I argue that it is 1/2. 

It is not as if I'm omitting the GB and/or BG, but rather, I'm combining them since order is not necessary/important in this scenario. BG = GB.  Making the original sample space (with their likeliness); { BB (25%), GB/BG (50%), GG(25%) }. And given the condition (one is a boy), the sample space then is reduced to {BG (25%), GB (25%)} as having a boy child is as likely as having a girl.

Is it an invalid assumption?
The sample space is not reduced to $\{BG (25\%), GB (25\%)\}$, the sample space is reduced to $\{BB (25\%), GB/BG (50\%)\}$, or equivalently to $\{BB (25\%), BG (25\%), GB (25\%)\}$. Reduced to this case, the probability of a girl as the other child is $2/3$.
How does the sex of the given child [boy] affect the probability of the 2nd child.  It doesn't.  It doesn't matter whether the first child is a boy  or a girl.  There are still only two possible outcomes for the sex of the unknown child:  B or G.  The probability is simply 1/2.  It's similar to this question:  A coin is flipped two times.  Given that the first flip is heads, what is the probability that the second flip is heads.  Thinking that what already happened will affect what happens next in events of chance is called the Gambler's Fallacy.  They could have 15 heads in a row, the probability of the 16th coin being tails is still 1/2.