Find probability of addition of random variables 
Suppose you toss a coin four times. The sample space Ω = {HHHH,HHHT,HHTH,...,TTTT} contains 16 outcomes and you should assume
  each outcome is equally likely. Let X be the Binomial random variable
  that corresponds to the number of heads in an outcome, e.g., X(HTHT) =
  2. Let Y be the Bernoulli random variable that evaluates to 1 if there is an even number of heads in the outcome, e.g., Y (HHHT) = 0 and Y
  (HTHT) = 1.
Let Z = X +Y, e.g., Z(HTHT) = X(HTHT)+Y(HTHT) = 2+1 = 3. What are the
  values of:
P(Z=0) P(Z=1)...P(Z=5)

Lets talk about Z=0. 
In order for Z=0 to be true, X=0 and Y=0. So then, to get P(Z=0), do I have to find the intersection of X=0 and Y=0 or do I add P(X=0) and P(Y=0)?

Remember $Y= \begin{cases} 0 &:& X\in \{1,3\} \\ 1 &:& X\in\{0,2,4\}\end{cases}$ so
$$Z=\begin{cases}1 &:& X\in\{0,1\}\\3 &:& X\in\{2,3\}\\ 5 &:& X\in\{4\}\end{cases}$$
