okay, random markets anyways don't exist, but just for the sake of discussion: I think your premise is wrong (for truly random markets).
If we are talking about truly random markets, then each (e.g.) 10 pip move up or down should always have a 50% chance. For example: price moves up 10 pips, then it moves up another 10 pips, then it moves down 10 pips.
Each 10 pip move here has a 50% chance of occuring, and is not dependent on the move that happened beforehand (if we talk about truly random markets).
For mean reversion strategies to work, our premise has to be that after a 10 pip move up, a 10 pip move down is more likely to occur than another 10 pip move upwards. But this is then not a random market at all.
The total % over this one month period was thus 54.3% that a 10 pip move would reverse 10 pips rather than continue another 10 pips in the same direction.
This leads me to believe that there is a slight advantage to a zero intelligence mean reversion strategy such as the one you describe above based on a 10 pip move. However, the probabilities change based on the pip movement amount chosen.