This question is vague, so the answer itself is unclear. At the risk of giving you more than you wanted, here're some answers anyway.
If you take "best from a probabilistic standpoint" to mean that the game-theoretic result is the best one possible, and all winning positions are equivalent (and all drawn positions are equivalent), then the answer is "no". That's because chess is almost surely a draw, so unless your opening move is so bad you lose immediately afterwards,* your first move is just a choice between different options that are equally good. In that sense 1. e4, 1. d4, 1. c4, and 1. b4 are all equivalent.
If on the other hand you take "best from a probabilistic standpoint" as the opening evaluation of the strongest chess engines today, then the best moves are likely 1. e4 and 1. d4. These are the two moves that engines are most likely to choose, if left to their own devices, on move 1. The exact "best move" depends on several factors however, such as 1) hardware used 2) time you let the engine think for and 3) which version of the engine is used (different versions of the same engine prefer different moves).
On yet another hand, if you take the "best from a probabilistic standpoint" as the move that wins the most in chess databases, then the best move depends on the database you use, how much importance you ascribe to statistical significance, and whether you impose any cutoffs like "master games only". Doing neither of the latter two means 1. Na3 is best according to these two databases. You write in the OP that "I'm trying to make no presumption on the skill level of players" - but if you take this literally, then worstfish** is a legitimate player, and you can easily skew your results by getting worstfish to play against random_move.
So what does "best from a probabilistic standpoint" mean to you? The answer will differ for different definitions.
*There is some circumstantial evidence that 1. g4 is the only first move that is losing for White.
**This is Stockfish configured to play the worst move available.