Probability using a defined sample space. Given a bag of $5$ different colour marbles: $R, G, B, W, Y$, we need to create a sample space to
study the outcome of when $3$ marbles are picked out of the five marbles in the bag.
The way I approached this, is that since we are creating a sample space for $3$ marbles picked
out of $5$, then there is $\binom53$ ways to make the selection.
Therefore, 
$$\begin{align*}
\binom53 &= 10\\
&= \{(RGB), (RGW), (RGY), (RBW), (RBY), (RWY), (GBW), (GBY),\\
&\qquad\qquad(GWY), (BWY)\}
\end{align*}$$
So now that I have my sample space how do I find the probability of a selection using my sample space which include for example $Y$. I know that without the sample space I can use combinatorics to do something like 
$$\frac{\binom42}{\binom53}$$
which gives the probability that the selection includes $Y$.

First let me point out a serious notational error: your sample space is fine, but it is a set of $3$-element sets of marbles, not a number, so it is not equal to $10$. Its cardinality is $10$.
The $10$ outcomes in your sample space are all equally likely, so each has probability $\frac1{10}$ of occurring. By actual count there are $6$ outcomes that include $Y$: $(RGY),(RBY),(RWY)$, $(GBY),(GWY)$, and $(BWY)$. Thus, the probability of getting one of them is $6\cdot\frac1{10}=\frac35$. Other problems can be done similarly: just count the outcomes in the event whose probability you wish to compute.
