How likely am I to beat a grandmaster through random play? Understatement of the week: Grandmasters are good at chess. I am not.

I was wondering because of this how likely it would be through complete random play to beat a grandmaster, and whether or not it was more likely for me to beat one through random play than out of my own skill set. (I am about 1400 FIDE, and therefore probably actively discount a lot of moves which would actually be quite good. 
So my questions are: 
1. How likely am I to beat a GM through random play?
2. Is this more likely than through my own skill?
The chance is absolutely zero. Not just against grandmasters, but against anybody.

In fact, it would be hard to win a game with random play against someone actively trying to lose, after all you would just as likely remove your pieces from the vicinity of the opponent's king than play the checkmating move, if the opponent finally managed to get to a position where mate is possible.

Everybody who knows the rules and has some sort of goal in mind plays much better than random play.