2016 AMC 12A 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\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.