AMC 12 2016a Problem 19

Jerry starts at $0$ on the real number line. He tosses a fair coin $8$ times. When he gets heads, he moves $1$ unit in the positive direction; when he gets tails, he moves $1$ unit in the negative direction. The probability that he reaches $4$ at some time during this process is $\frac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. What is $a + b?$ (For example, he succeeds if his sequence of tosses is $HTHHHHHH.$)

\textbf{(A)}\ 69 \textbf{(B)}\ 151 \textbf{(C)}\ 257 \textbf{(D)}\ 293 \textbf{(E)}\ 313