AMC 12 2019a Problem 20

Real numbers between $0$ and $1$, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is $0$ if the second flip is heads and $1$ if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x-y| > \tfrac{1}{2}$?

$\textbf{(A) } \frac{1}{3} \textbf{(B) } \frac{7}{16} \textbf{(C) } \frac{1}{2} \textbf{(D) } \frac{9}{16} \textbf{(E) } \frac{2}{3}$