lean_workbook_plus_29673

Now in order to find the expected number of moves before we hit our first barrier, we get the same equations as before,except $E_{-8}=0$ as well, so by symmetry, we know that $E_n=E_{-n}$ which makes the system much easier to solve, reducing it to the following equations: \begin{align*} E_0 &=\frac{E_{-1}+E_1}{2}+1=E_1+1\ E_1 &=\frac{7}{8}E_0+7\end{align*} and receive $E_{0}=64$ , meaning that the expected number of moves to hit the first barrier from zero is 64, so in total, the expected number of moves to hit both barriers is $64+240=\boxed{304}$ .