Random walk 1D with a single bound

Solution 1:

Short answer, the average number of steps does not exist.

To show this, suppose that the average (read: expected value) of the number of steps to move 1 step backward exists and is equal to $x$. We can make an equation for $x$ by multiplying the probability of each direction by the expected number of steps based on each direction. If we get a backwards step, then it only takes $1$ step. If we get a forward step, then we have $1$ step, plus the average number of steps to get back to the original position, plus the average number of steps to go backward. This becomes the equation:

$$x=\frac{1}{2}(1)+\frac{1}{2}(1+x+x)$$ $$\Rightarrow x=1+x$$

But this is a contradiction, so there is no finite average number of steps.