What is a probability model for the event "a random child is a girl"? An example in one of my book writes: A family has two children. You run into one of them and she is a girl. What is the probability that both children are girls given this information. Assuming that you are likely to run into either child, and that which one you run into has nothing to do with gender. 
The way the book solve this problem is this. Let $G_1, G_2, G_3$ be the events that the elder, younger, and random child is a girl, respectively. Then it proceeds to calculate $P(G_1\cap G_2|G_3)$.
I understand the book, but the only thing that really bothers me is the fact that I cannot find a probability model (a sample space, a probability function) to incorporate all $G_1, G_2$ and $G_3$. 
I tried $\Omega=\{GG, GB, BB, BG\}, G_1=\{GG, GB\}$ and $G_2=\{GG, BG\}$. Then there is no way to model the event "the random child$ is a girl with this space. 
Then I thought, oh, maybe I make another space to model that event, then form product space. Unfortunately, the result does not come out correct.
How would you do?  

The elements of your sample space need to include the information of which child you ran into.  Since you are equally likely to run the older child or the younger child (and this is independent of whether they are boys or girls), you can take your sample space to be $\Omega=\{GGY,GBY,BGY,BBY,GGO,GBO,BGO,BBO\}$, where $Y$ means you ran into the younger child and $O$ means you ran into the older child.  Each of the $8$ elements of $\Omega$ has equal probability.  The events in question are then $G_1=\{GGY,GBY,GGO,GBO\}$, $G_2=\{GGY,BGY,GGO,BGO\}$, and $G_3=\{GGY,GBY,GGO,BGO\}$.
In general, the elements of $\Omega$ need to encode all of the unknown (or "randomly determined") data.  Here that data includes not just the genders of each child, but also which child you ran into.
