Answer

This is a markov chain problem. You can tell because they ask for an expected number of steps until a result, and if you tried to evaluate all possibilities manually there are infinite paths you can take. You solve it by drawing a diagram and recognising when familiar states appear again.

Looking at the image above, we start off at state $S_0$, which is 3 edges away from the finish $S_3$. Next step invariably leads us to one of the three vertices which are 2 edges away from the finish (call it $S_1$). The key is to recognise that all these three states are equivalent - 1 step away from the start and 2 steps away from the end. In turn, expected number of steps from $S_0$ is $E_0 = 1 + E_1$ where $E_1$ is the expected number of states once we are in state $S_1$.

From there, we have $\frac{1}{3}$ chance of moving back to $S_1$ and $\frac{2}{3}$ chance of moving to state $S_2$ - which is two edges away from $S_0$ and one edge away from $S_3$. Finally, we have $\frac{1}{3}$ chance of moving to $S_3$ and $\frac{2}{3}$ chance of moving back to $S_2$.

Setting up the equations, we have: $$E_0 = 1+E_1$$ $$E_1 = \frac{1}{3} \cdot E_0 + \frac{2}{3} \cdot E_2 + 1$$ $$E_2 = \frac{1}{3} \cdot E_3 + \frac{2}{3} \cdot E_1 + 1$$ $$E_3=0$$ Solving the system, we find $E_0=10$