The exercise:

> 
> A box contains 5 white balls and 5 black balls. 3 balls are drawn randomly, one by one, without replacement. The following random variables are defined:
> 
> 
> $$ \begin{align}
> &\begin{aligned}
> X =
>  \begin{cases}
>  0, & \text{if the first ball extracted is white} \\
>  1, & \text{if the first ball extracted is black}
>  \end{cases}
> \end{aligned}
> \\
> &\begin{aligned}
> Y = \text{Number of black balls drawn}
> \end{aligned}
> \end{align}$$
> Find the conditional probability of $Y$ given $X$.
> 
> 
> 

$\textbf{My thoughts}$ about it:
So first of all, I check the possible values both random variables can take. For $X$ that is clear, and for $Y$ its possible values are on the set $\{0,1,2,3\}$.

Now I want to use the fact that: $\text{P}(Y=x|X=x)=\frac{\text{P}(X=x,Y=y)}{\text{P}(X=x)}$, for $x=0,1$ and $y=0,1,2,3$ to calculate what I'm being asked.

So, $\text{P}(X=x)=5/10=.5$ for either $x=1$ or $x=0$. After doing this I want to figure out $\text{P}(X=x,Y=y)$ for each possible value of both $x,y$. I think my sample space looks something like this
$$\Omega = \text { \{ (w,w,w),(w,w,b),(w,b,w),(b,w,w),(b,b,b),(b,b,w),(b,w,b),(w,b,b) \} }$$ where b stands for black marble, and w for white marble and their positions tells me order in which they were extracted.

$\textbf{My issue:}$ Is how to calculate any one of those probabilities, for example $\text{P(w,w,w)}$, since trying to calculate $ \begin{align}
&\begin{aligned} 
\text{P}(Y=0,X=1)=\text{P}(\text{0 black marbles extracted and 1st extracted marble is white}) = \text{P}(\text{Extracting 3 white marbles}) = \text{P(w,w,w)}.
\end{aligned}
\end{align}
$

Since there is no replacement there is no independence and I cannot make $\text{P(w,w,w)}=\text{P(w)P(w)P(w)}$ and I doubt the answer is $\frac{1}{8}$ since if I change the initial conditions to one million white marbles essentially my model would remain the same but clearly the probability of drawing 3 white marbles would increasy drastically.

Any insights or hints are much appreciated!
Thanks for reading!