Cumulative distribution function with 3 variables Let $X$ be the random variable whose cumulative distribution function is
$$
F_X (x) = \begin{cases}
    0, & \text{for} \space x\lt 0 \\
    \frac{1}{2}, & \text{for} \space 0\le x\le 1 \\
    1, & \text{for} \space x\gt 1 \\
\end{cases}.
$$
Let $Y$ be a random variable independent of $X$ and uniformly distributed over the interval $(0,1)$. Define the random variable $Z$ as
$$
Z = \begin {cases}
    X, & \text{if} \space X\le \frac{1}{2} \\
    Y, & \text{if} \space X\gt \frac{1}{2} \\
\end{cases}
$$
Determine $\mathbb{P} (Z\le \frac{1}{5})$.
I believe that $X$ only takes the discrete values $0$ and $1$ with equal probability, but I'm not entirely sure. By intuition, I think that the answer is $\frac{1}{2}$. I'm unsure about this question, so any advice would be appreciated.


I believe that $X$ only takes the discrete values $0$ and $1$ with equal probability, but I'm not entirely sure.

Yes, that is what the cumulative distribution function is saying: as it increases suddenly at $0$ and $1$, and remains constant elsewhere.  Thus $X$ has the probability mass function of:
$\qquad\mathsf P(X=x)=\begin{cases}1/2 &:& x=0~\text{or}~ x=1\\0&:&\text{elsewhere}\end{cases}$
So you should see that when $X=0$ then $Z=0$, or when $X=1$ then $Z=Y$, so: $$\mathsf P(Z\leq 1/5)=\mathsf P(X = 0)+\mathsf P(X=1\cap Y\leq 1/5)$$
