Funnily enough, there is some way to calculate this probability, and it's not vanishingly small. Reference: Elo World, a framework for benchmarking weak chess engines. This is more of a parody paper than anything, but the gist is that one can easily code weak chess engines and get them to play against strong ones millions of times.
To illustrate, some of the weak engines used are:
- random_move. Self-explanatory, and directly relevant to this question.
- same_color. When playing White, put pieces on White squares; when playing Black, put pieces on Black squares. If all pieces are already on the "right" color, make a random move.
- generous. Move so as to maximize the number of possible captures the opponent can make on our pieces. The more valuable the piece we're giving away, the more weight we give to that move.
- reverse_starting. "This player thinks that the board is upside-down, and as white, tries to put its pieces where black pieces start."
(I did say it's a parody paper, right?)
There're also a few serious engines. The strongest one playing is Stockfish1m. This is Stockfish, the strongest traditional chess engine, when asked to play a move after 1 million nodes (this means move after it has searched exactly 1 million positions). Exactly how strong this is I don't know. I know I can beat Stockfish 1 node fairly easily, but at 1 million nodes it's obviously a lot stronger, and it's likely strong enough to crush most humans.
If we assume that Stockfish1m is about as strong as a grandmaster, then there's an interesting table on the last page. Unsurprisingly there are many engines that are worse than random_move, and unsurprisingly Stockfish1m is the strongest engine. random_move is a full 2200 elo weaker than Stockfish1m, but it has a nonzero chance of winning a tournament between all these engines. In fact, p(random_move winning tournament) is 0.00000462. Small, but not zero.
Caveat: This isn't a direct head-to-head probability of random_move beating Stockfish1m.
Caveat #2: I am not completely sure where the line between the parody and the serious work is. It's possible the author is spoofing the use of Markov probabilities here, because p(random_move winning) = 0.00000462 sounds awfully large. Naively, I'd have expected something several orders of magnitude smaller.
Edit: I've convinced myself that the above is part of the satire. Here's a back-of-the-envelope calculation for the probability of random_move winning. If we assume that the grandmaster will never resign (and they presumably won't, since a random mover can always blunder a game) then the game might take 80+ moves to end in a mate, probably longer. Chess has a branching factor of about 35. Therefore, the random move needs to make "hit" roughly 35^(80) times in a row to win, making the odds around 10^(-123) against.