Conditional Distributions Choose a random integer $X$ from the interval
$[0, 4]$. Then choose a random integer $Y$ from the interval
$[0, x]$, where $x$ is the observed value of $X$. Make assumptions
about the marginal pmf $f_X(x)$ and the conditional
pmf $h(y | x)$ and compute $P(X + Y > 4)$.

I tried to make a sample space of the ordered pairs and came up with: 
$S=$ { $(0,0),(1,0),(1,1),(2,0),(2,1),(2,2),(3,0),(3,1),(3,2),(3,3),(4,0),(4,1),(4,2),(4,3),(4,4)$ }
(a total of 15 ordered pairs)
Next, I defined the pmf $f_X(x)$ as:
$f_X(x) =$
$\frac{1}{15} , x=0;\frac{2}{15} , x=1;\frac{3}{15} , x=2;\frac{4}{15} , x=3;\frac{5}{15} , x=4;$  
(sorry for the messy piecewise display)
My idea ended up being wrong. Can someone help me with this problem? 

If $X=3$, then we get sum $\gt 4$ with probability $\frac{2}{4}$, and if $X=4$ we get sum $\gt 4$ with probability $\frac{4}{5}$, so the required probability is $\frac{1}{5}\cdot \frac{2}{4}+\frac{1}{5}\cdot \frac{4}{5}$.
