Joint distribution of AB. Consider the following random experiment with two fair dice. First, the first die is thrown, and let the outcome be $A$. Then, the second die is thrown until the outcome has the same parity (even, odd) as $A$. Let this final outcome of the second die be $B$.
I am confused in how the joint distribution looks like. Could anyone shed some light? So, $A = {1, 2, 3, 4, 5, 6}$, but what is $B$? And then, what is $P_{AB}$?

We have various choices for the sample space. For instance, we may or may not be interested in the number of throws. It looks as if here we are not. So the outcomes can b chosen to be $(1,1)$, $(1,3)$, $(1,5)$, $(2,2)$, $(2,4)$, $(2,6)$, $(3,1)$, $(3,3)$, and so on. The total number of outcomes is $6\times 3$, and it is reasonably clear they are all equally likely. So each element of this sample space has probability $\frac{1}{18}$. 
Or else we could use as sample space all ordered pairs $(a,b)$ with $1\le a\le 6$, $1\le b\le 6$. But then we have to assign $0$ probability to $18$ of the ordered pairs, for example to $(4,3)$. 
