Occasionally when I'm bored, I'll play a game:

1. Pick a random positive integer $X$.
2. Add $+1$, $0$, $-1$ to make it divisible by $3$. $^\dagger$
3. Divide by $3$ to create a new $X$.
4. Repeat steps $2$ and $3$ until you reach $1$.

$^\dagger$ Keep track of how much you've added and subtracted. That is your "score".

Assuming a random starting number $X$ (rather than a pseudo-random one), is there an equal chance that this game results in a positive or negative score?

(I assume that this can be done with induction, hence the tag, but I'm fine with any proof)